Why do we read Yona of all things on Yom Kippur? There are better tales of teshuvah that are more relevant to klall yisroel; why was this sefer, which deals exclusively with non-Jews doing teshuva, chosen instead?
As it says in the Unetanneh Tokef, Yom Kippur is Judgment Day for all, not just for Jews:
וְכָל בָּאֵי עולָם יַעַבְרוּן לְפָנֶיךָ כִּבְנֵי מָרון. כְּבַקָּרַת רועֶה עֶדְרו. מַעֲבִיר צאנו תַּחַת שִׁבְטו .כֵּן תַּעֲבִיר וְתִסְפּר וְתִמְנֶה וְתִפְקד נֶפֶשׁ כָּל חָי. וְתַחְתּךְ קִצְבָה לְכָל בְּרִיּותֶיךָ. וְתִכְתּב אֶת גְּזַר דִּינָם:
בְּראשׁ הַשָּׁנָה יִכָּתֵבוּן וּבְיום צום כִּפּוּר יֵחָתֵמוּן
All mankind will pass before You like a flock of sheep. Like a shepherd pasturing his flock, making sheep pass under his staff, so shall You cause to pass, count, calculate, and consider the soul of all the living; and You shall apportion the destinies of all Your creatures and inscribe their verdict.
On Rosh Hashanah will be inscribed and on Yom Kippur will be sealed
Sefer Yonah reminds us that this is bigger deal than just the Jews. On the other hand, it's a kal v'chomer - If Nineveh does true teshuvah with such little impetus, we who have a brit with G-d should wake up and do teshuvah.
It also is a nice counterpoint to the haftarah for shacharit - Fasts aren't about sackcloth and ashes, says Isaiah (58:5) but about justice (ibid. v.6-7). You Jews are faking it. Look at the Ninevites instead, we hear in the afternoon. They can do real teshuvah. Don't assume that you are better at this just because you're Jews - you still need to work at it.
Also, what is another good sefer where all of a society does such a true teshuvah?
According to http://ohr.edu/1231, the Shlah says that their doing teshuvah inspires us to do it as well; and the Sefer HaTodaah ("The Book of Our Heritage") says that it's to show that you can't run away from G-d.
- To learn from the Teshuvah of the city of Ninveh (Siddur Rashi, Machzor Vitri, Rokeach). The Tzeidah Laderech adds: if the inhabitants of Ninveh who were not Jewish could fully repent, how much more so us who stood before Har Sina etc.
- To learn that one cannot flee from Hashem (Abudraham, Chofetz Chaim in Shaar Hatziyon 622:6)
- The fact that the ship was in danger of sinking because of an individuals wrongdoing teaches us that one can endure suffering for others. This is the idea of the death and suffering of tzadikim that atone for others, like Nadav and Avihu. (Chachmos Shlomo 622:2)
[Duplicated two reasons from other answers to provide sources].
R' Samson Raphael Hirsch says, in his extended commentary on Leviticus 16:10, that goats represent the God-given human power to resist, and that the Azazel goat used in the Yom Kippur service represents, specifically, the use of this power to (God forbid) selfishly resist God Himself. We have absolutely no use for this kind of resistance, so we send the goat that represents it far away from civilization, and allow it to perish in an ignoble tumble.
My son, upon hearing about this symbolism, suggested that Yonah, in trying to literally run away from God's directive to him, was acting very much in accordance with the trait symbolized by the Azazel goat. Perhaps his exiles into the sea and later outside of Nineveh echo what we do to the goat.