Jem and Scout are both exposed to incidents that most children would never experience at such a young age. Having a mysterious recluse like Boo Radley living next door seemed trivial after the other matters they witnessed that were related to the trial of Tom Robinson.
As most children do, Scout grew by leaps and bounds when she began attending school for the first time. Her connection with Dill gave Scout her first love interest, and she began receiving pressure to get rid of her overalls and begin acting like a lady.
After all, if Aunty could be a lady at a time like this, so could I. (Chapter 24)
She had also discovered that Boo Radley was a kind and gentle man.
"Mr. Tate was right."
Atticus disengaged himself and looked at me. "What do you mean?"
"Well, it'd sort of be like shootin' a mockingbird, wouldn't it?" (Chapter 30)
Jem was maturing in a different way, and by the beginning of Part Two we discover that he is experiencing the growing pains of puberty. He witnesses several examples of injustice (the jury's guilty verdict against Tom) and dishonesty (Nathan Radley) that he will never forget, but he also makes the conscious effort to be more like his father.
Jem was twelve. He was difficult to live with, inconsistent, moody... Overnight, it seemed, Jem had acquired an alien set of values... (Chapter 12)
Unlike the jury, Jem was wise enough to see that Tom was not a guilty man.
It was Jem's turn to cry...
"It ain't right, Atticus," said Jem.
"No, son, it's not right."