I think one lesson that Jem learns that profoundly affects him is two-fold: He begins to learn of the hypocrisy of the town and why Atticus stands in stark contrast to others in the town. As an adolescent, Jem is going through a process of change that is different from Scout’s. They both are exposed to the same events and gather some of the same lessons. I think one thing Scout learns on a fundamental level is to try to put herself in other people’s shoes; which she learns from Walter, Calpurnia, Tom Robinson and Boo Radley.
Jem is not as outwardly questioning as Scout but that doesn’t mean he isn’t as curious. Jem begins to understand a general sense of how a society functions, the social structures, class, race, and whom the society deems as outcasts and so on. He gets a lot of this by witnessing the bigotry in the wake of the trial. This is summed up perfectly in the last sentence of Chapter 23:
“I think I’m beginning to understand why Boo Radley’s stayed shut up in the house all this time . . . it’s because he wants to stay inside.”
This occurs right after Jem and Scout talk about how folks don’t get along. Jem said that if there were one kind of folks, everyone would get along. He seems to grasp that, like Scout said, there is just one kind of folks: Folks. And Jem seems to grasp that folks don’t get along for a pointless reason; because they are different.