Besides the basic facts of the story, you could ask questions that explore the development of the characters. For instance, when Tom Robinson's friends fill Atticus' kitchen with food the morning following Tom's conviction, what does this show about their feelings toward Atticus, and what does Atticus' reaction show about him as a man?
How does Miss Maudie help Jem understand the citizens of Maycomb? What does their conversation reveal about Maudie's character?
After discussing Tom's conviction with Atticus, how has Jem's view of people changed, and what does this change in attitude suggest about Jem?
Why would Harper Lee include the episode with the ladies of the Missionary Circle in the novel? What does this scene suggest about these women?
How did Maycomb react generally to the news of Tom's death and to Mr. Underwood's editorial? What does their reaction say about the white citizens of Maycomb?
Why does Jem react with such anger when Scout tries to tell him about Miss Gates? What's going on inside Jem's heart and mind?
Besides exploring the characters in the novel, these questions will lead you to some of its main themes.