So I would just like some insight on the reason for the commandments in the matter of whether they are for inheriting the world to come/heaven, or for they are for fixing this world. I have heard that they are for fixing this world and making the world a better place, but many christians assert that they are for inheriting heaven (as well with their belief in the idol). So if any of you have sources in anything that can help lead to the conclusion of either would be greatly appreciated.
Both of the reasons that are suggested in the OP are correct according to Torah. The primary reason is to inherit the World to Come, as explained in Mesilas Yesharim chapter 1:
והנה מה שהורונו חכמינו זכרונם לברכה הוא, שהאדם לא נברא אלא להתענג על ה' ולהנות מזיו שכינתו שזהו התענוג האמיתי והעידון הגדול מכל העידונים שיכולים להמצא. ומקום העידון הזה באמת הוא העולם הבא, כי הוא הנברא בהכנה המצטרכת לדבר הזה. אך הדרך כדי להגיע אל מחוז חפצנו זה, הוא זה העולם. והוא מה שאמרו זכרונם לברכה (אבות ד): העולם הזה דומה לפרוזדור בפני העולם הבא. והאמצעים המגיעים את האדם לתכלית הזה, הם המצוות אשר צונו עליהן האל יתברך שמו.
Behold, what our sages, of blessed memory, have taught us is that man was created solely to delight in G-d and to derive pleasure in the radiance of the Shechina (divine presence). For this is the true delight and the greatest pleasure that can possibly exist. The place of this pleasure is, in truth, in Olam Haba (the World to Come). For it was created expressly for this purpose. But the path to arrive at the "desired haven" (Ps. 107:30) of ours is this world. This is what our sages of blessed memory said: "this world is like a corridor before the World to Come" (Avot 4:16). The means that lead a person to this goal are the commandments which the blessed G-d commanded to us.
A secondary purpose of fulfilling the commandments is that the world as a whole is perfected by this. (Not necessarily in a physical sense, but by being used to fulfill God's will, He sees to it that the whole world is brought directly to its perfection.) As explained later in that chapter:
ואם הוא שולט בעצמו ונדבק בבוראו ומשתמש מן העולם רק להיות לו לסיוע לעבודת בוראו, הוא מתעלה והעולם עצמו מתעלה עמו. כי הנה עילוי גדול הוא לבריות כולם בהיותם משמשי האדם השלם המקודש בקדושתו יתברך
But if he rules over himself and clings to his Creator, and uses the world only as an aid to serve his Creator - then he elevates himself and elevates the world with him. For all creations are greatly elevated when they serve the "Adam HaShalem" (whole/perfect man) who is sanctified with the holiness of the blessed G-d.
These ideas are expanded upon throughout the first chapter of Mesilas Yesharim. I recommend studying it well.
There are numerous answers, and I cannot pretend to do a survey of all of them. I will just stay focused on the opinions that emerged in Eastern Europe since the Enlightenment turned "Why be Jewish?" into a question worth asking.
The Chassidic movement emphasizes G-d's Immanence. G-d's absence from the world is just an illusion. And correspondingly, a person's job in this world is to achieve deveiqus, an attachment to the Almighty. And then various schools of Chassidic thought emerged with differences of opinion not only in the details, but the relative importance of intellect, emotion and the experiential in how to achieve deveiqus with the Divine. And so you find that the Tanya, the book that gave articulatin to Chabad thought (primarily found in Lubavitcher chassidus) placed more emphasis on thought. As another example, Ruzhin emphasized the experiential, and thus the Chassidic court, the venue where the chassid has his most intense experiences. The descendent movements of Ruzhin still today treat their rebbe like royalty, investing in his clothing, home, nowadays his limo -- everything to create a feeling of grandeur which colors the encounters with G-d one has when visiting the rebbe.
When Chassidus started there was consequent pushback, and the Misnagdim (literally: Opponents) and other Jews primarily in Lithuania, articulated a worldview that was more directly a continuation of what had preexisted the new movement. The means to get "close" to G-d, who is not only Immanent but also Transcendent, is to emulate Him. Thus the goal of mitzvos is to achieve sheleimus (completeness) or temimus (wholeness with connotations of perfection).
So the two streams of answers to your question in that environment was that we do mitzvos to connect to the Almighty or to perfect our "tzelem E-lokim -- 'image' of G-d" by emulating Him.
Rav Chaim Volozhiner (early 19th cent Lithuania) instituted the modern yeshiva movement. (Although arguably today's yeshivos lack full continuity with the pre-WWII Lithuanian original.) He was a student of one of the central pillars among the Misnagdim, the Vlina Gaon. Rav Chaim wrote the notebooks that his son collected into Nefesh haChaim. Three of the four sections of Nefesh haChaim deal with mitzvot -- section 1 is about those of action, section 2 about prayer and blessings, and section 4 is about Torah study. (Section 3 is about how G-d created, and lays groundwork for section 4.)
According to Nefesh haChaim, through speech we repair the break between this universe and loftier existences. Qabbalah views this "break" as a necessary step in creating the universe; without some distance from the spiritual, the physical could not exist. But it is our job to unify all of creation.
In fact, being in the image of E-lohim, which means "Master of all Forces" means that humans alone are comprised of all the forces in existence. Animals work in the more physical realms, angels only in the more spiritual. Only human beings span these realms. And thus only humans can connect them.
For that matter, it is only the "lever" of the human soul by which actions in this universe can impact higher forces, because we are the only point of contact. And so, by doing mitzvot and refining one's soul one connects this world to the others, and thus perform "tiqun olam" as Qabbalah uses the term.
Similarly, section 2 discusses how speech brings the spirituality of the higher worlds into this one. By making a berakhah before eating an apple, one makes all the metaphysics that went into making that apple manifest. To eat without a blessing is robbery. Not, as you might assume, robbing the apple from G-d. Rather, robbing the world of the spirituality inherent in that apple.
And section 4 says that Torah study gives this world the "power" to continue existing. Because it is the inernalization of Divine Thought into the human mind, it brings G-dliness into the universe.
So in all three ways -- action, speech, and thought -- mitzvot act to metaphysically fix the world, but in the only way we can, by metaphysically fixing ourselves.
And much rests on the early chapters of section 4, and Rav Chaim Volozhiner's description of what it means to internalize the mind of G-d. Something that led to another ideological split, this one within Lithuania.
Rav Yitzchaq Volozhiner, the son who wove his father's works, added his own notes and introduction to produce Nefesh haChaim, wrote this memory of Rav Chaim in the introduction:
ה רגיל להוכיח אותי על שראה שאינני משתתף בצערא דאחרינא. וכה היה דברו אלי תמיד שזה כל האדם. לא לעצמו נברא רק להועיל לאחריני ככל אשר ימצא בכחו לעשות.
He regularly rebuked me, because he saw that I did not participate in the pain of others. And these were his constant words to me: This is the entire person. One is not created for himself, but to benefit others to the full extent of his abilities.
So all that fixing the world framework has to accomodate the fact that its author felt the whole purpose of a person is to "benefit others to the full extent of his abilities."
Within Lithuanian though there was another split. Less ideological than tactical. Both can be said to come from understandings of the early chapters of section 4 of Nefesh haChaim. He compares Torah study to immersion in the miqvah, it purifies the soul. And just as a person stays tahor after dry, a person's soul is shaped by the act of studying, even if the content is forgotten.
In what became the Yeshiva Movement, the belief was that one studies Torah and performs mitzvos, and the self-refinement happens in-and-of itself.
The Mussar Movement was founded by Rabbi Yisrael Salanter, who observed that this doesn't actually happen. Rather, Torah study of the sort the mitzvah calls for is the kind of study that is internalized and refines the soul. One has to consciously work at improving one's middos (character traits) to better imitate G-d and benefit others, and Torah and mitzvos are exercises Hashem gave us toward those ends.
My own philosophy is founded on Rabbi Shimon Shkops, a Lithuanian Rosh Yeshiva (in the city of Grodno) in the early 20th century who often spoke of his aspirations to emulate Rabbi Yisrael. Rabbi Shkop opens the introduction to his magnum opus, Shaarei Yosher, with the words (translation mine, taken from page 45 of my book, Widen Your Tent):
יתברך הבורא ויתעלה היוצר שבראנו בצלמו ובדמות תבניתו, וחיי עולם נטע בתוכנו, שיהיה אדיר חפצנו, להיטיב עם זולתנו, ליחיד ולרבים בהוה ובעתיד בדמות הבורא כביכול
Blessed shall be the Creator, and exalted shall be the Maker, Who created us in His “Image” and in the likeness of His “Structure”, and planted eternal life within us, so that our greatest desire should be to do good to others, to individuals and to the masses, now and in the future, in imitation of the Creator (as it were).
That phrase "planted eternal life within us" is taken from the berakhah before Torah study or the public reading. It refers to the Torah.
According to Rav Shimon, and of all the approaches the one I personally lean toward, the answer to your question is that we were given mitzvot so that we could (1) better connect to G-d so that we can share His Good with others, (2) develop the kind of character so that it is "our greatest desire", so that we are motivated to benefit others in the future.
The purpose of the Torah is human-oriented, not G-d-oriented since God is all-powerful and does not need to see people doing mitzvot, as if G-d needs people to observe the commands for some undisclosed reason.
In his Commentary on the Mishnah, Introduction to Perek Chelek Maimonides writes that people only achieve immortality when they develop their minds, not based on the care that they take in observing Torah commands. Thus, Maimonides stated that the purpose of the Torah is three-fold: to teach true ideas and helps improve individuals and society. People fulfill the Bible’s mandate, he stressed, when and only when they study and understand about science and the laws of nature and use their knowledge of the world to improve themselves and society. Thus, Maimonides considered it a mitzvah to study science.
"If G-d revealed the reasons for all the commandments, they would find ways to disobey them... In such a way, the entire Torah could be nullified... [However] there is not a single commandment that does not have a reason or a purpose. The majority of these causes and reasons, though, cannot be grasped or understood by the masses" (emphasis added).
We do not know the reasons for the commandments. We do not even know their relative importance. The Mishna says:
Be as scrupulous in performing a minor commandment as a major commandment, because you do not know the value of each commandment. [Pirkei Avot 2:1]
The Talmud adds:
The Holy One, Blessed be He, did not disclose the reward for the performance of commandments, so people should perform [all of] them faithfully [and not base their observance on each commandment's value]… You do not know from which of the commandments life [in the next world] will emerge for you. [Peah Y 7a]
The Talmud also gives the example of King Solomon. When he thought he knew the reasons for some commandments, he broke them, because he found a way to satisfy what he thought those reasons were, without observing the commandments proper:
Rabbi Yitzhaq also said: Why were the reasons of [most] Torah laws not revealed? Because in two verses reasons were revealed, and they caused the greatest in the world [King Solomon] to stumble. Thus it is written [in the Torah]:
[The king] shall not take too many wives, so his heart will not be turned away [from the Torah and towards idolatry]. [Deut. 17:17]
So [King] Solomon said, “I will take many wives and NOT let my heart be perverted.” Yet we read [in the Book of Kings]:
When Solomon was old, his wives turned away his heart. [1Kings 11:4.]
Again, it is written [in the Torah]:
[The king] shall not acquire too many horses [so as not to] cause the people to return to Egypt, [which is the greatest horse market.], because the Lord said: You shall not go back to Egypt. [Deut. 17:16.]
So [King] Solomon said, “I will acquire many [horses] and will NOT cause [Israel] to return [to Egypt].” Yet we read [in the Book of Kings]:
And Solomon had horses brought from Egypt… and a chariot came up and went out of Egypt for 600 shekels of silver. [1Kings 10:28-29] [Sanhedrin 21b]
Maimonides concludes his Sefer Ha-Mitzvot, the Book of Commandments, by admonishing us against this behavior:
This [story] is a lesson to people. If God revealed the reasons for all the commandments, they would find ways to disobey them. If even someone as great and perfect [as King Solomon] could make the mistake of thinking that he could do the forbidden act and avoid the underlying reason for the prohibition, how much more so the more weak-minded masses. Certainly [if they knew the reasons for the commandments] they would disregard them by saying, ““this was prohibited”, or “this was commanded” only for such-and-such a reason. I can avoid the reason for which the commandment was given and ignore [the commandment itself].” In such a way, the entire Torah could be nullified. God therefore concealed the rationale [for the commandments]. There is not a single commandment, however, that does not have a reason and a purpose. The majority of these causes and reasons, though, cannot be grasped or understood by the masses... [Rambam, end of Sefer Ha-Mitzvot]
Does that mean it’s wrong to look for reasons for the commandments? No. Rabbi Akiva tells us that it’s even a mitzvah to look for these reasons. [Eruvin 54b] However, in the end, no matter what conclusions we reach, we must continue to follow the commandments.