The two Finch children's loss of innocence actually begins before the story line of To Kill a Mockingbird takes place. Their mother died of a heart attack shortly after Scout's birth, and though Scout does not remember her, Jem does.
He remembered her clearly, and sometimes in the middle of a game he would sigh at length, then go off and play by himself... When he was like that, I knew better than to bother him.
The children's preoccupation with Boo Radley leads to fantasies about their creepy but unseen neighbor, but they eventually come to realize that the terrible neighborhood gossip is not true. Scout never totally gives up her hope of one day meeting him, and when she does, she discovers that he is a gentle man who heroically saves their lives from the murderous Bob Ewell.
Jem is affected by Nathan Radley's cementing of the secret knothole, and the lie that Boo's brother gives for the reason. Jem recognizes it as a hateful act meant only to further isolate Boo from the world outside. The death of Mrs. Dubose also saddens Jem, and he has to deal with the mixed emotions of her actions, which he discovers were primarily as a result of her long addiction to morphine. The jury's verdict in the Tom Robinson trial troubles Jem more than it does Scout--Jem seems to understand the facts of the case better than Scout--and he loses faith in the townspeople as well as in the lawful purpose of juries.
"I always thought Maycomb folks were the best folks in the world, least that's what they seemed like."
"... Can't any Christian judges and juries make up for heathen juries..."
Scout discovers that her teachers are not always the best educators. She recognizes Miss Caroline's limitations on the very first day, and she sees the hypocrisy in Miss Gates' beliefs when she defends the Jews in Germany but denounces the Negroes who live in Maycomb. Scout also realizes that many churchgoers (i.e. the missionary circle) have similar beliefs.
Jem and Scout are both exposed to the underpriveleged side of Maycomb; some are poor but well-meaning, like the Cunninghams, while others make no attempt to improve themselves (the Ewells). They also get a first-hand look at the evil that exists--first with the lynch mob, and later at the hands of Bob Ewell. But they also find that people sometimes get what they deserve: Bob Ewell's death makes Maycomb a better place, and Sheriff Tate's refusal to name Boo as Bob's killer is a case of justice best served.