The narrator of To Kill a Mockingbird by Harper Lee is Scout Finch, and though it is clear from the beginning that she is writing from the perspective of time, she is only six years old when the story begins. Jem is four years older, and the story covers the next three years of their lives. Of course there are dramatic changes in the thinking and behaviors of any children during those crucial years, but these siblings have more to cope with than most, and naturally they learn many life lessons from those experiences.
One of the things they learn through the course of the novel is to consider things from other people's perspective. Atticus says:
You never really understand a person until you consider things from his point of view...until you climb into his skin and walk around in it.
One way the kids learn this lesson through their experiences with Boo Radley as they mature from making fun of him and assuming that all the rumors are true to having an appreciation for someone who is obviously not able or willing to be part of "normal" society. They learn this lesson slowly, but they do learn it. From Boo's thoughtful and observant gifts to his repairing of Jem's overalls and finally to his saving their lives, both Jem and Scout learn that Boo is a real person and they come to respect him.
The kids also grow in their understanding of what courage is. Jem whines that his father is too old to play football and Scout is unimpressed with her father's excellent lawyering abilities. They do not appreciate their father's good character and his willingness to stand for what is right until they see him in action at Tom Robinson's trial. Like most kids, they think about courage in terms of physical ability, which is why they are impressed when they watch Atticus take down a rabid dog with a rifle. What they learn is that courage takes many forms, and they learn this lesson from their experience with Mrs. Henry Lafayette Dubose. Her courage in kicking her morphine addiction before she died is most impressive to Jem, but both of the children learn the lesson.
One other lesson the children learns comes from something Atticus says to them when they receive their air rifles for Christmas. Atticus delicers his famous mockingbird line, and neither child seems to really understand it at the time. Scout asks Miss Maudie about it:
“Your father’s right,” she said. “Mockingbirds don’t do one thing but make music for us to enjoy...but sing their hearts out for us. That’s why it’s a sin to kill a mockingbird.”
Of course that helps, but still Scout and Jem do not quite understand that this actual mandate is also a metaphor for the kinds of people who might be considered mockingbirds in life. Through their experiences with Tom Robinson and Boo Radley, it is clear that both kids come to understand this metaphor. Jem gets the idea most through Tom's trial, as he is more aware than Scout of the injustice that has been done. It is appalling to him, which is a sure sign of his maturity. Scout understands this truth most through her experience with Boo Radley at the end of the novel. In fact, she is the one who specifically makes the connection between Boo and a mockingbird. Understanding this idea as a literal as well as a figurative truth is an exceptional example of both Jem and Scout's maturity over the course of the novel.
Obviously kids learn and grow over time, but Jem and Scout learn a few specific and significant lessons throughout the course of this novel.