In Chapter 11, Jem and Scout go to church with Calpurnia because Atticus is out of town (at the state legislature) and Calpurnia isn't sure their teacher will be at church that Sunday. When they get there, a woman named Lula asks why Cal is bringin "white chillun to a nigger church." Cal shuts her down with "It's the same God, ain't it?" After services, they return to find that Aunt Alexandra has moved in with them. In Chapter 14, the news that they'd been to church with Cal and that Cal had invited Scout to her own home came out in conversation, and Scout says, "Atticus, I'll go next Sunday if it's all right, can I?"
Aunt Alexandra responds, "You may not."
The exchanges at church and with the Finches demonstrate the racist attitudes on both sides. Some of the blacks don't want any whites mixing with them "in their church" and Aunt Alexandra doesn't like the idea of Atticus's children mixes with blacks any more than necessary.
Another important scene occurs in Chapter 15, when Tom Robinson is moved to the Maycomb jail. Atticus walks to the jail to keep watch and Scout, Jem, and Dill follow him, not understanding what is going on. A lynch mob appears and Atticus--with Scout's help--manages to convince them to go back home. The fact that they are a lynch mob and mean to break into the jail and torture Tom Robinson to death (which is what lynching is) is not explicitly stated, but that's the gist of what is happening here.
Another scene that strikes me as important to clarifying the racist attitudes of the townsfolk is the point where Dill gets sick to his stomach during the trial because "Old Mr. Gilmer" was "doin' him thataway, talkin' so hateful to him"--"him" being Tom Robinson. Scout and Dill leave the courtroom, and Mr. Dolphus Raymond, considered the town drunk, offers Dill a sip of something out of a paper bag "that'll settle your stomach" (Chapter 20). Dill discovers that Raymond only has Coke in the bag, and asks why. He explains that "some folks don't--like the way I live" (he's married to a black woman), so he tries to "give 'em a reason." He explains that he lives the way he lives, being married to a black woman and "having mixed children" because that's the way he wants to live, but it's easier for everyone if he can give them a reason to believe he's "in the clutches of whiskey" and just can't help it.