Since your question is divided into two parts - rabbinical responses and how you personally can respond, I'll include in my answer two parts as well: my own personal advice and rabbinical responses and responses of other individuals that I've gathered over time.
First of all, a little about me: I've been interested in Tanach study, Jewish history and archeology since I was little, but only in the last couple of years did I start dedicating significant portions of my time to researching these subjects of Biblical Criticism, the Documentary Hypothesis, the Biblical Minimalism/Maximalism debate, Jewish chronology vs world chronology, Jewish ties to neighboring cultures during the Biblical and Talmudic eras, the impact of archeological discoveries on our understanding of Tanach, and more. I'm far from an expert on any of these, but I think I may know a little more than the average person.
To be able to even begin to discuss these subjects, really the only thing you can do is research, b'chinat: "Know what you shall answer an apikores" or even "know thine enemy". This can include:
Reading books on Biblical archeology, academic works on Tanach, news articles on new discoveries1, essays, the original Documentary Hypothesis texts and so forth.
Listening to lectures on said subjects. Many academic or academic-level lectures are available online.
Visiting museums and archeology sites that are relevant to the subjects that you're researching - the former can give you a clear picture of the artifacts and the latter will give you some understanding of the geography and geology of the subject matter.
What I can do here is give you a bit of an outline of the some of the more well-known subjects and some pointers when attempting to approach the subject:
Biblical Criticism, it may be argued, was started by our sages millennia ago. Every discrepancy pointed out in the text is criticism of the text, but when we're talking about Biblical Criticism, it's usually a reference to the secular criticism which started more or less with Baruch Spinoza and Thomas Hobbes. Spinoza, famously, in his Tractatus Theologico-Politicus quotes the Ibn Ezra on Devarim 1:2 who spoke about "the secret of the twelve" - twelve verses in the Torah which according to him weren't written by Moshe but by later people (we're familiar with the question about the authorship of the last eight verses of the Torah, but Ibn Ezra began expanding the number of "non-Mosaic verses"). Spinoza used this as evidence that the Torah wasn't written by Moshe but by a much later author, perhaps Ezra. As such, it makes the text meaningless because it wasn't handed down to Am Yisrael at Mt. Sinai by Hashem and is simply a judicial document that might have been relevant at one point, but no more.
Spinoza's view paved the way for the well-known late-18th century-mid-19th century "Documentary Hypothesis", in its original form: The Graf-Wellhausen Hypothesis. The theory expounded on Spinoza's and other previous academic research and suggested that not only wasn't the Torah written by Moshe, but it was actually a combination of four (later, some opined it to be five) different documents authored by different people or groups of people in the ancient Israelite culture and eventually all the texts were combined into one. The documents were titled J (or Y), E, D, P (and later also DH). The basis for this theory are the various textual discrepancies that appear in the Torah. For example, verses that it wouldn't make sense for them to have been written by Moshe, such as the last eight verses or the list of Edomite kings in Beresheet. Another example is the changing of Hashem's name from the Shem Hameforash in one chapter to Elokim in another, which may point to a theological difference between different Israelite cults.
Though many view Wellhausen's work as that of an objective secular academic, in fact, Wellhausen was something of an extremist Christian and an antisemite. Wellhausen proudly ended his work Prolegomena zur Geschichte Israels by pointing out how he has now disproven the Israelite religion and brought glory to JC.2
The Documentary Hypothesis in its original form is now no longer considered valid as a number of the critiques were proven to be based on false assumptions, so the DH was altered and modified over the years. Unfortunately, the vast majority of secular academics still take the DH in one form or another to be factually correct.
To break apart the secular criticism, there are a few approaches one can take:
Taking the classic Torah view on the matter. For example, the explanation to why there are different names of Hashem in the Torah is not because there were different religious factions among the ancient Israelites but because Hashem governs the world through different ways and methods. Rabbi Mordechai Breur is famous for his view on the need to accept the Biblical Criticism as-is but to face it head-on with a Torah-based approach. His position on the matter came to be known as "Shitat Habechinot" ("the aspects approach"). Some of his essays in Hebrew can be found here. I also recommend his book שיטת הבחינות של הרב מרדכי ברויאר which is a compilation of several of his essays on his approach, replies by other scholars and his own replies to the replies.
Another rabbi who faced the criticism head-on (but with a different approach) is Rabbi David Tzvi Hoffman (Radatz Hoffman), in his commentary on the Torah. Especially recommended is his work on Vaykira. Some of his books can be found here.
Showing that the criticism is based on false assumptions, especially in how Biblical linguistics worked. Rabbi Prof. Moshe David (Umberto) Cassuto took this approach in his work (though he claimed that his work was from a purely academic viewpoint). I recommend checking out his The Documentary Hypothesis and the Composition of the Pentateuch: Eight Lectures by U. Cassuto (called in Hebrew תורת התעודות) (the introduction and lecture eight are available online here). His more famous works are his commentaries on Beresheet and Shemot (I personally have yet to read them).
A lesser-known figure is someone whose work I've quoted in quite a few answers I've written here on MY in recent months, Rabbi Ahron Marcus. I discovered his work last summer. Unfortunately, many of his books were lost during the Holocaust, but those that we do have are a real treasure trove of information. In the context of Biblical Criticism, most central is his book Barzilai, a work on the psychology and origins of the Hebrew language (originally written in German and translated into Hebrew by one of his grandchildren)3. At the same time, Rabbi Marcus refers to the works of many of the leading scholars of his time in the field of Biblical Criticism and Biblical archeology, breaks down their arguments against the Tanach and offers his own views on the matters. The book is available online in German here and in Hebrew might be found in some used bookstores or in Otzar Hachochmah.
A great book that outlines many of the different views on various Biblical Criticism topics and suggests some solutions is Rabbi Amnon Bazak's Until This Day.
A good lecture to hear is Rabbi Professor Shnayer Leiman's class on Biblical Canonicity.
Note: Nowadays, Biblical Criticism is divided into two categories: Higher Criticism - everything that has to do with the concept of the DH, and Lower Criticism - everything that has to do with textual contradictions.
Biblical Archeology (Minimalism/Maximalism):
Biblical Archeology refers to archeological digs and studies of Israel and neighboring countries mentioned in the Tanach with regards to the eras described in the Tanach. For many years, it was a given that the majority of the Tanach was factually correct and this premise was the basis of many archeologists' approach during their work. As such, when a new discovery was made, the first question was: How does this fit with the Bible? An interesting anecdote is told in The Romance of The Last Crusade, by Vivian Gilbert, a British officer during WWI, who tells the story of another officer who figured out where the pass that Yonatan used during the Battle of Michmas was and recycled Yonatan's strategy in order to defeat the Turkish garrison of that area. This approach is now known as Biblical Maximalism. Some of the more famous scholars and researchers in this field include Kenneth Kitchen, Amichai Mazar, Yigal Yadin and Adam Zertal.
Around the 1970's a new approach to Biblical Archeology arose in Europe and quickly spread around the world, becoming most prominent by the 90's: Biblical Minimalism. As outlined in the Wiki page:
Biblical minimalism...is a movement or trend in biblical scholarship...with two main claims:
that the Bible cannot be considered reliable evidence for what had happened in ancient Israel;
that "Israel" itself is a problematic subject for historical study.
The basic idea of Minimalism is that the Tanach is unreliable because its authors weren't contemporaries of the eras they wrote about4 and as such, had no idea what they were talking about, didn't understand the past cultures, the geographical differences, the religious evolution of the people, the linguistics and so forth. A classic example, now long debunked but still unfortunately used by some of the lesser-knowledgeable followers of this school of thought, is that the camel was domesticated much later than the time of the Avot, so for Avraham to have had camels "must be" a later invention, inserted into the story.5 Then evidence was found of much earlier camel domestication and the theory was debunked. Another example is the doubt of the existence of King David, until a fragment of a stele was found in Israel with the phrase "Beit David" inscribed on it. Currently, though, this is the only extra-Biblical evidence of David, so Minimalists still have a leg to stand on and can claim anything from the name being an invention of one of the Davidic kings to King David being the ruler of a small tribal state and not of a great, unified Israelite kingdom. Some of the more famous scholars and researchers in this field include Yisrael Finkelstein, Niels Peter Lemche and Kathleen Kenyon (though she was apparently less outspoken on this subject (and also passed away long before Minimalism really drew hype)).
Note: In more recent years, some of the scholars on both sides of the division have attempted a kind of reconciliation by attempting to find common ground in their views and making some concessions.
An excellent book to read is Dr. Yitzchak Meitlis's Excavating the Bible (לחפור את המקרא in Hebrew). Meitlis, a religious archeologist and lecturer, is on the Maximalist side of the debate.6 Despite having been a student of, among others, Yisrael Finkelstein, the well-known Minimalist, he isn't afraid to criticize their research in his book, but also gives their due when he finds them to be correct.
Another good book is Rabbi Ahron Marcus's Kadmoniyot (also available on Otzar Hachochmah), a book dedicated to analyzing many archeological discoveries of his era (late 19th-early 20th century) and showing how they line up with the Tanach.
I also recommend taking a look at this list on Wikipedia of artifacts relating to this field.
One of the most famous debates on the originalism of the Tanach based on archeological evidence is on the story of the Flood and Noach's Ark. In the late 19th century, an archeologist named George Smith discovered in Nineveh the library of King Ashurbanipal and in it, among other things, an ancient work titled The Epic of Gilgamesh. After deciphering the eleventh tablet of the text, he concluded that it was the story of a Mesopotamian flood and saw this as evidence that the Biblical Flood was a copy of this story with a monotheistic twist. An excellent essay refuting all of Smith's arguments and of those that came after him was written by Rabbi Dr. Izaak Rapaport and can be found here (pg. 27-35) (or my Hebrew translation with added background information (though still going through the editing process) here). A different approach was taken by Rabbi Philip Bieberfeld in Universal Jewish History, vol I (available here), who attempted a more amalgamated solution6. In fact, most of the people that I've heard speak about the subject usually go with the approach that the fact that there are so many cultures around the world that have flood stories is evidence that the story is true. For example, Rabbi Yehoshua Zitron's class on the topic.
Another is the seemingly lack of evidence for the Exodus. This is a very complicated issue, perhaps more so than most other Minimalist/Maximalist debates because there are disagreements even among the Maximalists over which pharaohs were the ones mentioned in the Torah and some artifacts that can be considered evidence for the Exodus relate to one pharaoh while others relate to a different pharaoh. From what I've seen, most agree that Ramesses II "The Great" and his father Seti I are the Exodus pharaohs. However, there are historians who argue for different pharaohs. For example, Rabbi Alexander Hool in Pharaoh: Biblical History, Egypt and the Missing Millennium argues for Pharaoh Thutmose II. I haven't read his book yet, but a short outline of Rabbi Hool's argument can be found in this review by Rabbi Reuven Chaim Klein. Another example is Damien Mackey from the University of Sidney, who argues in several essays (all can be found here; for example, this) that the entire Egyptian chronology must be reorganized, including removing several pharaohs who actually appear twice or more, or weren't really pharaohs but high-ranking officials in other dynasties. In his view, the pharaoh of the Exodus was Neferhotep I.
Pieces of evidence for the Exodus can include: Comparisons between the Torah and what we know about Ramesses II, a relief from the Book of Gates that may detail the Egyptian take on the Splitting of Yam Suf, the Tempest Stele and the Ipuwer Papyrus. On the Minimalist side of things, views can go anywhere from there was no Exodus to there was a small Exodus of only the Levites.
A third example, relatively less well-known, is the discovery of a Neo-Assyrian text telling of the mythic birth of King Sargon of Akkad:
"...My high priestess mother conceived me, in secret she bore me. She set me in a basket of rushes, with bitumen she sealed my lid. She cast me into the river which rose over me. The river bore me up and carried me to Akki, the drawer of water. Akki, the drawer of water, took me as his son and reared me..."
As can be clearly seen, there are similarities to the story of Moshe. However, most people nowadays who hold the DH to be true agree that the Torah was written prior to the dating of this text, which is the 7th century BCE. Therefore, it's much more likely that the people who wrote this myth based it on Moshe and not the other way around. An analysis of the two stories can be seen here and some calculations on the matter can be seen here.
Jewish Chronology VS World Chronology:
A complicated issue in itself is the question of how the traditional Jewish chronology fits with the world chronology (or vice-versa, depending how you look at it). The main problem is that the traditional Jewish view of history is primarily based on the chronology presented in Seder Olam Rabbah. There are a few problems with the SOR chronology. One is that it doesn't feature all of the Persian kings known to history nor does it present a length of time for the Persian Empire that could theoretically contain all of the kings. Another problem is that it says that that Bayit Rishon stood for 410 years and Bayit Sheni for 420, but other records and calculations make out the two Temples for existing for longer.
Good overviews of the issue can be found in the Wikipedia page on this issue, this post in the Hebrew blog ארץ העברים, in this post on sabbah hillel's blog and in Rabbi Anthony Manning's classes on the subject: Old version, new version pt. 1, new version pt. 2
Over the years there have been many who have offered various solutions for the discrepancies between the two chronologies. Mitchell First brings a vast majority of the various opinions in his book Jewish History In Conflict (parts of it can be read here). Rabbi Alexander Hool offers his own theory in The Challenge of Jewish History: The Bible, the Greeks and the Missing 168 Years.
The gist of the various opinions is that someone lied. The question is: Who, why and to what extent? Some say the Persians lied. Some say the Greek historians lied. Some say the Christians lied. And some even say that Jewish rabbinical authorities lied. Some say even Daniel was in on it.8 The one exception is the view that the world chronology is correct and that SOR was never meant to be taken literally but is more like a collection of midrashim. As such, even among Jewish Biblical Maximalists, there are many that view SOR as non-authoritative and follow the world chronology, such as Rabbi Mordechai Breur, Rabbi Ahron Marcus, Professor Yaakov Liwer, Professor Yehoshua Meir Grintz and Dr. Yitzchak Meitlis. In other words, to choose to accept the world chronology is not necessarily problematic.
To conclude: Information is important. The more information you have on these various subjects, that better you can hold your ground when discussing these subjects. Much as it may seem that the secular approach has the high ground - that's not actually a simple case to make, when looking at all of the facts. But one has to know the facts first.
1 Around Purim-time I spoke with a guy I knew from yeshiva who is now starting his third year as an archeology student here in Israel. He informed me that a significant portion of discoveries made aren't revealed to the public until long after the initial find. That's why a news header can be the click-baity "AMAZING NEW 2ND TEMPLE PERIOD DISCOVERY MADE" and when you actually read the article, it'll say that the artifact was actually found anywhere between half a year to a few years prior. So...not exactly new. To get in on the know of these things you'd either have to be part of the research field or subscribe to archeology magazines like the BAR.
2 Wellhausen claims in his writings that the original works of the Tanach were the Nevi'im and that the Israelite culture was based off of the connection the prophets had with God, until "the rabbis" came and forced the Torah portion of the Tanach onto the people of Israel, until JC came and revealed the truth to the people. Forget, of course, that JC kept most of the Torah.
3 Unfortunately the second half of the book, which, among other things, was said to discuss the meanings of Tanachic names, was never published and the manuscript was lost during the Holocaust.
4 Which is partially true in some cases. For example, Yirmiyahu was only a contemporary of the last few kings of Judea, yet wrote about everything from the final days of David to the end of the Judean kingdom in Melachim.
5 In The Bible As History by Werner Keller, written before the discovery of evidence of early camel domestication, he suggests that "camels" are actually "donkeys".
6 I've heard that being religious - whether Jewish or Christian - doesn't make it a given that one would be a Maximalist. Werner Keller, mentioned above, was Christian but decidedly minimalistic in his views, at least at the time that he wrote The Bible As History. Certainly, many non-religious scholars can be found on either side of the debate. Adam Zertal z"l used to say that his research couldn't be considered to be based on religious subjectivity because he was one of the most secular researchers out there, having grown up in an extremely left-wing, secular kibbutz in Israel.
7 I highly recommend checking out Rabbi Bieberfeld's footnotes and looking up the sources he uses. Some can even be found online. A particularly useful site for finding such texts is archive.com.
8 Note that there are some nafka minot to arriving at the conclusion that our sages were the ones that falsified history. For example, it means that the Hebrew year is actually much later, much closer to the year 6000 (!). Another is that it would mean that our count of Shmittin and Yovelot is wrong. As told by Rabbi Manning, the latter reason is what ultimately made Rabbi Shimon Schwab go back on his conclusion that the discrepancies were created by our sages (for Rabbi Schwab's entire back-and-forth on the matter, see here).