You'll find a fairly long list of questions for your consideration about this book in the Topics for Further Study section of the enotes study guide:
In addition to these questions, you might add some simpler or more direct ones. For example, what does it mean to focus on racism this way?
Why would Wright include two female characters with the same first name? (Is the confusion related to the concerns about race in the novel?) What about the nicknames in the novel? What does their use tell us?
And of course, to cut to the heart of things, ask questions like, "How much does this work still apply? What does it mean as race relations change?"