I always understood Judaism as being the most peaceful religion, the one that doesn't go on crusades or various campaigns of forced conversions. However, how do I approach the fact that we killed the nations in Canaan in order to get Eretz Yısrael? Because at the end of the day our doing so is no different than any other religion claiming they kill because of their religion.
First, we can't take that much credit for not launching a Crusade or Jihad -- for the past 2,000 years we lacked the opportunity! So, as a debating point with outsiders, it won't get you very far.
That said, within Judaism, there aren't the grounds for doing so. We do not believe in a world where everyone is supposed to be Jewish, or that non-Jews are fated for a horrible afterlife, or any some such.
So, what about the Canaanites...
Hashem plunked us down in the middle of a horrible barbarian society. Where girls like Rachav were routinely placed into prostitution by their own family (!) when she was 10 (Zevachim 116b) and boys were routinely burnt alive as sacrifices to Molekh. (Later ritualized into passing the son over the fire, or between two fires.)
Still, there are limitations on the war. We must offer peace first (Devarim 20:10-11, also Sifri, Devarim #199, and codified by the Rambam at H. Melakhim 6:1). Which means non-combatants were given a chance to escape, if their rulers would allow the option. When laying siege to a city, you have to give the civilians a route to escape. (Ramban on Sefer haMitzvos, assei #5, Rambam ibid 6:7) Also, women and children are not to be killed. (Rambam ibid 6:4; although one may kill someone posing an immediate threat, even if a child, as a danger to life would trump this halakhah.)
What does this gain, if the law is genocidal? Let's get back to the Crusades for a moment. No, we weren't expected to tell the Canaanites "convert or die". But, those escaping could choose to observe the 7 mitzvos benei Noach and leave the label "Kenaani" (Canaanite) that way. For a deeply polytheistic people, it is a significant conversion of sorts.
None of which directly answers your question, just modified it. It's where I run out of sources.
Here's my own opinion: Since we were, indeed, dealing with a bunch of barbarian warriors, let's understand what would happen if we showed any weakness. It could be that we were told to take out Amaleiq entirely, people and culture, the way new prisoners in fiction are advised that to get peace, they should start by taking on "the toughest prisoner in the cell block". And similarly a strong stance with the other local warring tribes. What looks to us, living in a modern civilization, as a violent path. may well have been advised by G-d as the path to the least death overall!
But we didn't actually finish either job -- neither killing out Amaleiq nor dispossessing the Kenaani tribes. Hashem then brings civilization to the region -- through Assyria and Sanhcheirev. An empire that sought unity by mixing populations, weakening loyalty to one's own specific tribe and land at the expense of Greater Assyria. The "Ten Lost Tribes" are lost at this point, but the identity of the Amaleiqim and Kenaanim are also lost. All of these laws become moot. And just when the force of a civil government makes my suggested reason for such laws also moot.
I will only address the aspect of why God commanded killing the Canaanite nations, as I gather that is the basis of your question.
Devarim (Deuteronomy) 27:16-18, states that regarding the "close nations", namely, the Canaanite nations, you should not offer them any peace treaty but rather kill every soul. Verse 18 explains why - so that they should not teach you to do the abominations that they did to their gods.
The land of Israel is unlike other lands in the world. Everywhere else, people inhabit the land and act as they wish. When God promised the land of Israel, he wanted the land to be a holy place inhabited by holy people who would obey God's laws as given to B'nai Yisra'el. We see numerous times throughout the Torah that God specifically states that one of the punishments for not following his laws would be exile.
A good example of my point can be found near the end of Vayikra (Lev.) chapter 18. It lists illict relationships, and then mentions not to sacrifice your child to Molech, and some other items. Beginning in verse 24, its says not to defile yourselves with all these things, because this is what the other nations did. These people defiled the land, and the land purged them. (Actually, G-d caused the land to do this.)
Verse 28 is the "key" verse. It says not to imitate the nations habits so that the land won't purge you in the same way that it purged the other nations.
So, in short, it is as if the land of Israel has an inherent "personality". Things such as idolatry, harlotry, illicit relationships, child sacrifices to gods etc. cause the land itself to be "sickened" and "vomit" its inhabitants. Thus, G-d doesn't want His chosen people to be influenced in any way by such inhabitants. Call it "preventive medicine", given by the ultimate doctor (See Ex. 15 where God is called a "doctor".)
As for your claim, "it is no different than any other religion claiming they kill because of their religion," I am not knowledgeable about the reasoning or foundations of every other world religion to know exactly why they murder people. If you are able to show that in their religious book (such as the Koran, etc.) that God commanded killing people with an explanation of exactly when and why that command was given, we can have an intelligent discussion / debate on this. Beyond that, I have demonstrated that God's commandment to destroy specific nations as above as well as Amalek, was limited to that time, only, and only those people. Other than self defense, there is no instance where God has commanded killing a nation. So, without a different religion explaining these same ideas, I think we can conclude that another religion claiming killing for religious purposes is a man-made idea and not God-given.
It is important to note that, before starting war, there is a commandment (obligation) to offer peace. According to at least Ramban and Rambam that also applied to Canaanite nations.
When you approach a town to attack it, you shall offer it terms of peace (Devarim 20:10)
On this the Ramban comments (translation artscroll)
[...] the calling out for peace must be done even in an obligatory war, for we are required to call out for peace even to the seven Canaanite nations [as can be seen with Sihon (2:26), one of the seven nations].
Ramban continues and explains one Canaanite city did indeed make peace with Israel (the Hiviite in Gibeon) and comments all is asked of them is to accept the seven Noahide commandments.
The Rambam codifies this in MT Melachim 6:1
War is not conducted against anyone in the world until they are first offered peace (and refuse it), whether this is a Discretionary War or a War of Mitzvoh, as it says, “when you come close to the city to fight with it, you shall call to it to make peace” (Deut. 20:10). If they make peace and accept the Seven Commandments incumbent upon the Sons of Noah (Gentiles), none of them are killed [...]
and in 6:4
If they do not come to peaceful terms or they make peace but do not accept the Seven Commandments, we engage in war against them [and bad things happen]
I'll try to address the question of "why us to do the job" instead of "ה' ילחם לכם ואתם תחרישון" as is customary to the "דרך העבודה" approach (if you're not familiar with this approach, please don't downvote it before checking it out). I would also prefer to refer to those nations as if they were theoretical to ease the understanding, because, maybe, they were.
There's evil in this world (no kidding) and not only the "Yetzer", but also there are people (and the whole nations) that were created evil, just as there are nations that were created good (ישראל עלו במחשבה תחילה).
Undoubtedly, the purpose of our existence in this world as "the good guys" is to fight "the bad guys" - the bad inclination, the bad nations etc. That could be done in two ways: 1. amplifying an perpetuating the "light of the Torah" by engaging in studying it, and thus "pushing the evil off" from a distance. 2. by fighting it "in contact" in the physical world by performing physical Mitzvot, marrying women, raising kids, working and also judging and executing the verdicts.
Both ways are legit (see my answer Why did Moshe send these spies?), however the later is thought to be more efficient albeit more dangerous. However, the preferred way must fit the qualities of the generation (the dispute between Moses and the Spies) - there are generations (Dor Hamidbar) where the first way is appropriate, and generations (Dor HaAretz) where the second one is.
That time was very special in opportunity to exterminate the evil by Dor HaAretz lead by Moses himself, which would lead to immediate establishment of the Mikdash forever, which could not be destroyed. The Spies saw it differently, fearing the danger of learning the bad misdeeds from the evil nations instead of conquering them, as the Torah warns us all the time. That decision created a huge "mess" in the following history as we know, the extermination of the evil nation was never final, so were the two Temples.
This Mitzvah was explicitly given to the generation that could distinguish the people of the nations. That skill was long lost, just as the practical commandment of wiping those nations off and remains dormant till the coming of the Moshiah B"B.
Your idea is excellent. Only people who think about what they read will catch this contradiction. It's true that it seems as though the Jews held a crusade. However, I offer another approach.
The conquest of Canaan
Any person with some kind of sensibility would agree that the Spanish conquest of the Americans was a genocide. Where the Nazis ultimately failed in the holocaust (with the reestablishment of Israel), the Spaniards succeeded, genocide worked. There are no Aztecs alive today. They were either killed by gunpowder or diseases such as smallpox. This aside, Deuteronomy 7:2 seems to say that Jews must mistreat Canaanites, saying:
“You must smite them, utterly destroy them; make no covenant with them, and don’t be kind to them.”
The Torah spoke to a nation on the brink of war. The Torah invites the Israelites to smash their idols and slaughter women and children and yet, Moshe changed this rule and sought peace, a higher goal in the Torah. Of what right did Moshe change this Torah law? Maimonides writes that G-d had to allow certain practices as concessions to human nature. Thus, the Torah allows the laws of slavery even though G-d does not want the practice of slavery to continue, for it is an abomination to G-d. After all, He led the Israelites out of slavery, how much more so should He abolish the practice altogether. This is the time to enforce such a law. But G-d does not abolish slavery. Instead, the Torah modifies these laws, implying that they should be modified even further. For example, while it allows sacrifices, it restricts them to certain animals, likewise, while it allows slavery it prescribes the legislation and treatment of slaves as humanly as possible, so much so, that the rabbis later said,
“he who acquires a slave acquires a master over himself.”
Rabban Yochanan ben Zakkai understood the true message of the Torah. He greeted everyone, even pagans, with blessings of peace (Berakhot 17a). Rabban Gamliel shared bread with a stranger (Eruvin 64b). Jews gave charity to non-Jews (Tosefta Gittin 3:13). Jew even visited non-Jews when they were sick (Shulchan Aruch, Yoreh Deah 151:12). The Torah also stresses the need to treat non-Israelites fairly, for we too were once slaves in a foreign land. Thus, the Torah’s ultimate goal is to drive people away from pagan practices. The rabbis recognized the Torah’s intention and did away with sacrifices in 70 C.E., (the destruction of the temple was a good time to stop). About this time, they also abolished slavery. These changes are in accordance with what the Torah desires.
War and Peace
In his Mishneh Torah, Hilkhot Malakhim, chapters 5 and 6, Maimonides says that the Israelites must make peace with the Canaanites if they are able to do so.
Deuteronomy 20:10 states:
“when you approach a city to wage war against it, you must propose a peaceful settlement.”
If the Canaanites agree to the peace and observe the basic laws of civilization (the Seven Noahide Laws,) the Israelites must accept them with peace. Maimonides explains that this was Joshua's first move when he entered Canaan. Three letters were sent, allowing them to flee, make peace, or wage war. A lot chose war, especially in the case of the Gibeonites.
Maimonides says that the Israelites were not required to kill any Canaanite offering peace. And when the Torah does say that the Israelites killed all the inhabitants, it was, to quote Rabbi Ishmael, “speaking in human language,” in other words, the Bible always exaggerates to highlight its message. There are no exceptions. The statements should not be taken literally any more than the tower of Babel “reached into the heaven.” In any event, the Israelites were defending themselves against unprovoked attacks from belligerent nations.
In his Mishneh Torah, Concerning Kings and Wars 5:1 and 6:1, 4, Maimonides states that Jews should never harm Amalek during a time of war without offering a ceasefire under certain circumstances.
Returning to Joshua, it follows that the command to eradicate the Canaanites was a temporary injunction. The nation needed to hear a call to arms at that time. Nevertheless, Joshua never conquered nor destroyed the inhabitants of Canaan.
 The Aztecs were not deserving of such a fate due to the sacrifices. It was a horrendous act, one which the Spaniards should have tried to influence, but not kill, pillage, rape, and enslave. Also, the Torah bans human sacrifice
 Babylonian Talmud, Kiddushin 22.
 (Leviticus 25:35)
 The Babylonian Talmud, Sanhedrin 96b, says that not all of the Amalekites should have been killed. For many Amalek repented and became proselytes to Judaism, studying Torah in B’nei Berak
 There is little archaeological evidence to support a mass-invasion of Canaan. It follows that the Hebrews settled peacefully, except for the few exaggerated reports in the Bible