Yonah arose to flee to Tarshish from before God... (1:3)
We read Sefer Yonah on Yom Kippur because it mentions not only the repentance of the sailors and Ninevians, but also the repentance of Yonah. God gives him a task, yet he flees from that task, as quickly as he can to as far away as possible. Of course, fleeing from your assigned task does not work. After three days in the fish, Yonah realizes that his only option is to flee back to God, through his prayer, to the task which has been assigned to him.
It is in our nature to flee from our assigned tasks in life. Hopefully we can realize the futility of this flight before reaching the metaphorical "belly of the fish", the lowest and most degraded point imaginable. But in the event that we have fled this far, we must realize that only one further destination remains, and that is to flee back to God.
"And the end of all our exploring
Will be to arrive where we started
And know the place for the first time." (T.S. Eliot)
(cf. R' Baruch Gigi, 9/15/07)
"Should I not have pity on Nineveh, that great city, in which there are more than 120000 people who cannot discern between their right and left hands, and many domestic animals?" (4:11)
The weird part of this rhetorical question is clearly the mention of animals. How can they be equated with human beings in terms of God's attention and mercy?
There is one previous mention of domestic animals in Sefer Yonah. When Yonah first came to Nineveh and announced that the city would be destroyed, the people decided to repent. They fasted and put on sackcloth - and made their cattle and sheep fast and put on sackcloth as well. (3:7-8) It seems that God's final comment to Yonah, mentioning the people of Nineveh along with their animals, is an allusion to the people's previous choice to include the animals in the fast.
Now, thinking that your animals too must fast is not exactly an example of spiritual sophistication. It seems that the Ninevians do not fully comprehend the nature of moral autonomy and free choice, which humans possess and cattle do not. As God says, they "cannot discern between their right and left hands" - they are spiritually clueless. And yet, given the opportunity, they leave their evil deeds and choose to repent.
We, who are better educated and recognize that cattle are cattle and humans are human beings, can learn from the example of Nineveh. If even these spiritually clueless people can and do make the choice to repent, then what about us?