One of the important ways that Harper Lee chose to show prejudice and persecution throughout the entire novel was by allowing the overall story about Tom Robinson and his trial for rape show one form of prejudice and persecution and then Scout's own life and her reaction to the events to show another.
The entire story is built around the fact that the town's prejudice against Tom Robinson because he is black and that this prejudice will lead to his persecution and eventual death because of a wrongful conviction. Atticus' work to defend Tom serves to bring out the likelihood of his innocence but it is unable to overcome the prejudices held by the jury.
But Scout struggles to deal with her own issues around prejudice involving Boo Radley, Walter Cunningham and the Ewells. She is taught a lesson when Walter Cunningham comes over for dinner and she is rude about the fact that he pours syrup all over his food. Her father and Calpurnia help to straighten out her prejudice against the young man based mostly on the fact that he doesn't have nice things and has different manners. She learns that he is a hard-working, kind and decent young man, helped to this realization by her father's treatment of him as an equal when he has dinner with them.
There are many other instances of different people's prejudices and the resulting persecution which serve to demonstrate their ubiquity.