Choosing just two overarching frames for each, Scout learns to consider the world from the perspective of others and Jem learns about social classes and that hypocrisy does exist in the adult world. These are just two progressions. There are other themes of progression one could trace through the development of either character.
One of the first times Scout is taught to consider someone else's perspective is when she is reprimanded for making fun of Walter Cunningham, Jr. when he is eating with her family. Calpurnia tells Scout that the Finch family might be "better" (wealthier) than the Cunninghams, but treating Walter so poorly suggests that Scout is not better (in the ethical sense):
"Yo‘ folks might be better’n the Cunninghams but it don’t count for nothin’ the way you’re disgracin‘ ’em—if you can’t act fit to eat at the table you can just set here and eat in the kitchen!”
Scout will later come to Walter's defense when Aunt Alexandra calls him "trash" and forbids Scout from inviting Walter to her house.
Scout had also been quick to make fun of Boo Radley. After lessons from Miss Maudie, Atticus, and others, Scout begins to see this lesson of considering other people's perspectives. After the attack, Scout walks Arthur (Boo) home and, literally and figuratively, "sees" things from his perspective:
Atticus was right. One time he said you never really know a man until you stand in his shoes and walk around in them. Just standing on the Radley porch was enough.
One of the most telling moments in Jem's increasing understanding of the hypocrisy and class distinction in Maycomb occurs at the end of Chapter 23. Jem confirms to Scout that Walter is not trash. Then Jem gives his analysis of the kinds of people in town. It is a simplistic analysis but it conforms to the way most people in Maycomb think:
"There’s the ordinary kind like us and the neighbors, there’s the kind like the Cunninghams out in the woods, the kind like the Ewells down at the dump, and the Negroes.”
Jem boils it down to the fact that certain people have been reading and writing longer. Scout says there aren't four kinds of folks; there is one kind of folks. "Folks." Jem says he used to think this way but the adults don't get along. Considering this problem, Jem concludes that Boo Radley stays inside his house to avoid these kinds of conflicting social divisions.
For similar reasons, Jem takes Tom's conviction harder than Scout or Dill. Having watched the trial, Jem believed Atticus had won the case. When Tom is found guilty, Jem is more upset at the injustice of the verdict than he is that his father lost. This shows a sign of maturity in that Jem learns an unfortunate truth about the adult world: that the courtroom is not immune to prejudice.