At the beginning of the summer, Jem is a boy who likes adventure and seeks to satisfy his curiosity about Boo Radley's reclusiveness. He is quick to make up stories about Boo, and equally quick to play games to feed those stories. Gradually he begins to suspect that Boo is not the monster he, Dill, and Scout made him to be to suit their games during the summer. Finding the dolls and other gifts in the tree causes him to suspect the tenderness in Jem. Apart from this, he likes to give advice to his sister and tease her about being a girl, which makes him feel like a boy and brother. The trial has a signficant effect on Jem, and when Atticus loses he is deeply hurt, losing faith in the goodness of people and power of the law. These recede into the background as he continues to mature, and as he does he has less time for his sister. We see him becoming more "of a man" as he protects his sister as the story works toward its climax. He walks her to the performance at school, and late at night he puts his hand on her to help walk her home. then, he fights Ewell to save his sister, having his arm broken in the process. By the end of the story he is this young man, ready to take on some of the responsibilities we see his father shouldering throughout the story.
One point in the novel where we see a change in Jem is when Atticus shoots the rabid dog. Scout immediately wants to go tell everyone about it, but Jem tells her not to. He seems to come to some kind of understanding about Atticus during those moments. He also makes a remark at that point about being a gentleman like Atticus.
In the course of the novel, Jem leaves childhood behind and becomes a young adult. Initially, Jem is a fun-loving child. Along with Scout and Dill, he concocts plays and participates in all kinds of foolish dares. Later he and Scout build a snowman. He tries to smuggle a note into Boo. All of these are done as a child. However, once the trial looms, Jem's innocence fades. He is exposed to the cruel nature of Maycomb society. Physically, he is becoming a man. He shows Scout a hair sprouting on his chest. Emotionally, he is trying to act more responsible. He even goes so far as to threaten to spank Scout if she doesn't behave, and he informs Atticus when Dill runs away from home and ends up hiding under Scout's bed. After the fallout from the trail, Jem truly leaves every last shred of childish innocence behind. That is one reason he gets so angry at Scout when, late in the novel, she brings the trial up again. As a young man, he found a way to get over the injustice of the verdict. Being reminded of that, reminds him too of his naive beliefs and actions.