I assume that you picked up on how the story of the trial of Tom Robinson offers a commentary about racism and how various people join in with it, or oppose it (openly or quietly).
I also assume you noticed that with the Boo Radley plotline, TKAM addresses the matter of people who don't fit in to society, the rumors and fear that surround them, and how they may in fact be gentle souls.
Here are a few things I didn't fully appreciate the first time I read the story.
Childhood. One reason TKAM is so entertaining is that it is told from the point of view of a young child. Much of the fun in the story comes from Scout's descriptions of the games, legends, and value system that she, Jem, and Dill share as children. Also, her description of the first day of kindergarten is not to be missed.
Much of the humor in the story comes from the fact that Scout, though she is the narrator, doesn't understand what is going on in certain scenes. One notable one is the scene where Jem lies and tells his father that the kids have been playing strip poker.
Scout's innocence and cluelessness become poignant in the scene where Atticus is guarding the jail, and Scout barges in to the middle of a hostile mob and begins chattering away.
Coming-of-age. TKAM can be read as a coming of age story, not about Scout, but about her older brother Jem. In the course of the book, Jem goes from being a nine-year-old firmly planted in childhood, to a moody almost-13-year-old who is coming to grips with the world around him. Early in the book, he figures out that Boo Radley has been reaching out to him and Scout, though Scout does not realize this.
Later, we see Jem lose his innocence when he is sure that Tom Robinson will be acquitted, but instead the jury find him guilty. The morning after the trial, Miss Maudie debriefs this shock with Jem, and she "promotes" him to the adult world by giving him, for the first time, a full-sized piece of cake instead of a small cake in a patty-pan. Atticus also talks Jem through his dismay that the law could allow such an unjust thing to happen.
Often, we have to read between the lines to see the struggles and changes that Jem is going through, because these are described from Scout's point of view.
Southern values and culture. Though TKAM offers a devastating condemnation of racism in the culture of the 1930s South, it does not portray the culture as wholly bad. We see the racism in the trial, and the snobbery in Aunt Alexandra and her circle of ladies. We see the gossip, personified in Miss Stephanie Crawford. But we also see some good Southern values in Miss Maudie, who "had an acid tongue in her head," but was unfailingly kind to the children, was an avid gardener, and who faced the burning down of her house and yard without even a hint of dismay. We see dignity, formality and never-failing politeness in many adult characters, but especially in Atticus. And we see the horrible old Mrs. Dubose, who is a snob and a racist but who is determined to break her morphine addiction before she dies. Atticus calls her "the bravest person I ever knew."
Parenting. Read through the book, paying attention to all the good things Atticus does as a parent. Notice how he expects the children to respect him, Calpurnia, and other adults (such as Aunt Alexandra), even when he does not agree with them. Yet he has unique parenting values that do not match perfectly with the values of his family and society, and sometimes he has to fight a bit to pass these on to his children (e.g. when Aunt Alexandra comes to live with them). Notice too the subtle ways he tries to prepare the children for the jeers they will face when Tom Robinson comes to trial.
In a sense, everything Atticus does is for the sake of living with integrity in front of his children. Atticus sends Jem to read to Mrs. Dubose so Jem can see an example of bravery. He is a crack shot, but he shoots a gun exactly once in the book, because he does not want his children to learn that violence is something to be proud of. At the end of the book, Atticus can barely stand to cover up the way that Bob Ewell died:
"Sometimes I think I'm a total failure as a parent, but I'm all they've got. ... if I connived at something like this, frankly I couldn't meet [Jem's] eye, and the day I can't do that, I'll know I've lost him. I don't want to lose him and Scout, because they're all I've got."