One example is Mrs. Dubose's comments to Scout and Jem as they pass her porch. She is obviously unhappy that Atticus has chosen to defend Tom Robinson, and shows her racism when she says things like
"Your father's no better than the niggers and trash he works for!"
She is demonstrating her racism in her commentary, and her attitude towards Atticus. She also obviously thinks that black people rank in the same category as "trash."
Another example is when the group of angry men congregate outside of the jail to do harm to Tom Robinson. They want to take him out, hurt him, or possibly worse, simply for being black. Then, there is the delightful Mrs. Merriweather and Mrs. Farrow, who prattles out their racism at the missionary circle that Aunt Alexandra hosts. They talk about how their black servants have been acting downright "sulkym dissatisfied" because of the trial, and how it was no good to "stir 'em up," referring to the black people in town. Mrs. Farrow says that it's no good trying to educate or better black people...it is just "a losing battle".
One other example is Bob Ewell who harrasses Tom's wife after the trial, trying to intimdate her into not going to work, and not passing his direction in her walk. He, as a white man, thinks that he is better than her, so he bugs her and scares her, and thinks that he has the right to do so. Also, consider Adolphus Raymond, a white man who chooses to live with a black woman. He is the scandal of the town, and must carry around fake liquor in order for people to think that he is drunk, and thus excuse his "crazy" behavior. The town's racism comes into play as the refuse to accdpt that a man in his right mind would choose to spend time with a black woman.
I hope that those examples help a bit; good luck!