This is an excellent question while at the same time being perhaps too large to really answer in just a few paragraphs. What you are really asking about is one of the main mitzvot that we practice, namely davening with intention (בכוונה או בעיון), meaning in plain translation with the meaning of the words. See Mishnah Peah 1:1 and Shabbat 127a.
If you are fortunate, this goal will be one of your endeavors for the rest of your life. You success to that end will only be limited by your proficiency in Hebrew and Aramaic.
An excellent source to learn the meaning of words in prayer is the Siddur Kol Yaacov. You can read it online at the following link:
If you find it helpful, it can also be ordered in hardcover from the same link.
Chen (חן), Chesed (חסד) and Rachamim (רחמים) are loosely translated as Grace, Kindness and Mercy. They are the words used to identify the three pillars of the system of existence which the Creator established in all of creation. Left is for "Chen", right is for "Chesed" and the middle pillar is for "Rachamim". Everything is based upon it.
It is the basis for the 3 Avot, Avraham (who is associated with the quality of Chesed), Yitzchok (who is associated with the quality of Chen or Gevurah) and Yaacov (who is associated with the quality of Rachamim). This idea is also found in the opening blessing of the Amidah and also in the opening paragraph of Birkat HaMazon.
This is also the source of the idea that the nation of Israel is "Am Segulah" which many translate as "Treasured Nation" or "Chosen Nation". But is actually better understood as "Adaptable Nation". Segulah also means adaptable. It derives from the vowel point called "Segol" which is 3 dots.
In the same fashion, on Rosh HaShanah and Yom Kippur, we say in the "Unatanah Tokef Prayer" (ונתנה תוקף) which emphasizes the 3 pillars of Torah, Tefillah and Tzedakah by which any harsh decree is overturned. A link for this prayer can be found at the following link:
This idea is also implied from the first three letters of the Aleph-Beit, namely "Aleph", "Beit" and "Gimmel". The "Aleph" has a meaning of learning and teaching which corresponds with Torah. For more details see the following link:
And this leads to the letter "Beit" to be associated with the pillar of judgement and "Gevurah". Just like the Torah begins with the letter "Beit" in "Bereshit" which talks about the beginning of creation. And in so doing, created the concept of duality which is a concealment of G-d's oneness. And yet at the same time it is also associated with the concept of blessing (ברכה). Ultimately, this concealment of HaShem's unity will be revealed for good and blessing. This is also understood from the paragraph of "Kadesh" (Shemot 13:1) said at the end of morning prayers in connection with tefillin. It emphasizes that we are to remember that G-d answered our prayers and brought us out of Egypt with a strong hand (כי בחזק יד כו״). This "strong hand" is explained in the Haggadah to be associated with the plague of "dever" which is judgement and severity. This is how it appears externally. Yet remembering this in connection with the hand tefillin which is placed on the weak hand, the left arm, emphasizes to us that the "strong hand" is the right hand, the side of kindness. The aspect of judgement is concealed kindness and actually saves us.
Chen (חן) is usually understood to be an abbreviation for the phrase Chachmat Nistar (חכמת נסתר). This is often translated as "Wisdom of Concealed Things" borrowing from the meaning as it is applied to Queen Esther's name (see Megillah 13a). But it also has a connotation of "Protection" as found in Midrash Tehillim on Psalm 19:7. This idea is also mentioned by Rabbi Moshe Cordevero in Pardes Rimonim, Sha'ar Archei HaKinuim in connection with the word "Chanun" (חנון). He explains that this is the idea of kindness even though one doesn't merit kindness. It comes from the posuk, "And He said: I will make all My goodness pass before you...and I will be gracious to whom I will be gracious and will show mercy to whom I will be merciful." (Shemot 33:19) Thus the letter "Beit" and the idea of "Chen" are associated with the aspect of tefillah.
The letter "Gimmel" is associated with the third pillar, "Rachamim" and the concept of "Gemilut Chesed", to do acts of kindness. This is the concept of "Tzedakah". It also has a connotation of strength, like "Gevurah" but from a different direction. In this case, it is associated with the idea of specifically where the hand tefillin is placed, the "Tigboret" (תגבורת), the high spot of the muscle in the arm. This is associated with the idea of having the strength to lend assistance to those in need. And through this strength one joins and unifies the other two pillars. For more details, see the following link:
It is important to keep in mind that there is a process which begins in Elul and extends all the way through to Shemini Atzeret and Simchat Torah. Elul up to erev Rosh HaShanah is the time when we prepare for Tishrei. We try to get our "books in order" through "Cheshban HaNefesh" and if something is lacking or incorrect through excess, balance the checkbook. That balancing is the extra acts of kindness, Torah study and prayer that we do during the month of Elul. G-d is accessible and "in the field" so to speak. Available to everyone no matter the circumstance.
On erev Rosh HaShanah, going into Rosh HaShanah, G-d relinquishes His Kingship in a manner of speaking. This period is called the "Nesirah" (נסירה) from the root (יסר) meaning "removed" or "restricted". The purpose of this is so that we can renew our acceptance of G-d as our King. This is derived from the posuk, "There is no King without a people." (אין מלך בלא עם)
We announce our acceptance of His Kingship and crown G-d King through the blowing of the shofar on Rosh HaShanah. And the following days are manifesting that Kingship in progressively greater levels until it is manifest in the most physical and mundane things during Sukkot.
According to Sefer Meori Ohr and Sefer Pardes Rimonim: Sha'ar Archei HaKinuim, "Hashiveinu" (השיבנו) and "Karveinu" (קרבינו) are closely related concepts. The root of Hashiveinu is "Teshuvah" (תשובה) which in turn is from "Shabbat" (שבת). This root of Shabbat (ש׳ בת) is comprised of the letter "Shin" and the word "Bat", meaning daughter. Daughter alludes to the concept of Malchut (Kingship) which is a central theme of Rosh HaShanah. The form of the letter "Shin" is made up from 3 "kavim", 3 lines or "Vavim". This indicates that the precursors needed from us prior to the revelation of G-d's Kingship are the 3 services of Torah, Tefillah and Acts of Kindness. And these correspond with the 3 general systems which G-d used to create the universe, namely the right kav which is "Chesed", the left kav which is "Gevurah" and the middle kav which is "Rachamim" or "Tiferet" which mediates between the two extremes and inclines all toward kindness in an equitable manner.
In a similar fashion, "Karveinu" is associated with both "Karev" (קרב) which means to draw close together and also has an implication of that this drawing together is because of what is within. That we all have one "source" and "father" (אב אחד). It is also associated with "Korban" (קרבן) which is the offerings made in the Temple. These were the vehicle by which the "sparks" were elevated, something done now with blessings and prayers. "Korban" (קרבן) has a value of 352 (שנ״ב) which parallels the concept for "Shabbat". Only this time it is (ש׳ בן) a "Shin" followed by "Ben", "son". The aspect that we are all children to HaShem
(בנים אתם להוי׳). And this is the running theme of "Avinu Malkeinu" (אבינו מלכנו), our Father, our King.
The word "Salach" (סלח) has a value of 98. The Meori Ohr points out that this is the value of G-d's 4 letter name (הוי׳) together with the expanded form of that name called "Av" (ע״ב). In that form of G-d's name the "Yud" is (יו״ד). The two "Hehs" are like this (ה״י). And the letter "Vav" is expanded like this (וי״ו). The concept of name expansion is the idea of how His blessings manifest into existence from above existence, meaning from His transcendent 4 letter name. In the case of this particular name it is associated with "Chesed", kindness (חסד) because His kindness is eternal (כי לעולם חסדו). "Salach" is also associated with "Segulah" (סגלה which has the same value of 98) which means both treasured and adaptable as discussed above.
The Meori Ohr says that "Chamal" (חמל), "Chalam" (חלם), "Melach" (מלח) and "Machal" (מחל) all come from the same root. Their value is 78 (ע״ב) which is 3 times G-d's name (3 times 26). This again relates back to the concept of making motzi at the Shabbat table and dipping the bread לחם) 3) times in the salt(מלח). This repeats the idea of the 3 kavin mentioned above and how this results in everything being inclined toward kindness. And this is what we declare in benching with "And He gives bread to all flesh because His kindness is eternal."
(והוא נותן לים לכל בשר כי לעולם חסדו)
Unfortunately, that is all I have time for today. Shabbat Shalom!
May you be written and sealed for a good and sweet year both spiritually and physically.