It was not only colored people who praised John, since they could not, John felt, in any case really know; but white people also said it, in fact had said it first and said it still. (1.1.29)
John pleases everyone around him, white and black. It's interesting that the narrator points out that white people praised him first. Although it's not clear that that's what the narrator is saying, it is clear that John's relationship to white people is very different from his father's.
"You come here, boy," he said, "and see what them white folks done done to your brother." (1.1.159)
Gabriel uses Roy's accident as an object lesson, not about what trouble John might get into if he runs with the wrong crowd, but rather about what might happen if he gets too near white people. He sees the wound as an insult from one race to another, the only interpretation he can give it.
"You see?" came now from his father. "It was white folks, some of them white folks you like so much that tried to cut your brother's throat." (1.1.165)
This line kind of makes it hard to trust Gabriel. He says that the white people tried to cut Roy's throat… when his wound is up on his forehead. But remember that there was another Roy, whose throat was slit in Chicago, who was John's half-brother.
In details such as this he was always very easy—he would always put on a clean shirt, or get his hair cut, or come with her to Uplift meetings where they heard speeches by prominent N****es about the future and duties of the N**** race. (2.1.73)
For Florence, it's very important to be ambitious and to try to improve one's situation. When her husband behaves himself, obeying her, it's all in matters of appearance. His clothes and hair are, for her, signs that he is trying to better himself. Going to Uplift meetings were another way she convinced herself he was a good guy.
"Where you expect us to live, honey, where we ain't going to be with n*****s?" (2.1.98)
Unfortunately, all of Florence's ambitions fall on deaf ears with Frank. He doesn't see any way out of their situation, so he doesn't see any reason to try. Florence wants to live among educated people, not with slackers like Frank's friends. Her use of the perjorative "n*****" refers to black people with a lack of ambition.
"I thought I married a man with some get up and go to him, who didn't just want to stay on the bottom all his life!"
"And what you want me to do, Florence? You want me to turn white?" (2.1.103-04)
Unlike Florence, Frank doesn't believe in any kind of racial uplift ideology. For him, the way the world is makes it impossible for him to move up; he would have to be white, which is impossible.
"You ain't got to be white to have some self-respect! You reckon I slave in this house like I do so you and them common n*****s can sit here every afternoon throwing ashes all over the floor?" (2.1.106)
The problem with Florence's reaction to Frank's lack of ambition is that she ends up reinforcing the racism she is trying to fight against. When she uses a racial slur she forgets that she is included in it by the white people around her.
There was silence; he whistled again a few bars of his song; and then he yawned, and said: "Is you coming to bed, old lady? Don't know why you keep wasting all your time and my money on them old skin whiteners. You as black now as you was the day you was born."
"You wasn't there the day I was born. And I know you don't want a coal-black woman." But she rose from the mirror, and moved toward the bed. (2.1.141-42)
Florence's bedtime ritual reveals how race affects beauty standards. She uses whiteners to make her dark skin lighter, and even though Frank tells her she doesn't need to, she is sure that he would prefer a lighter-skinned woman to a dark-skinned one.
[…] "I just mean you better be careful, son. Ain't nothing but white folks in town today. They done killed… last night…"
"I know," he said abruptly, "but they ain't going to bother me. They done got their n***** for this week. […]" (2.2.318-21)
When Gabriel bumps into Royal in town, he is terrified for the boy's life. A group of white people murdered and tortured a black soldier the night before, and so most of the black people in the community are hiding out at home. Royal is cocky, though, treating the murder as a common, weekly occurrence that he can't fight.
"Well, maybe, we go to a museum."
The first time he suggested this, she demanded, in panic, if they would be allowed to enter.
"Sure, they let n*****s in," Richard said. "Ain't we got to be educated, too—to live with the motherf***ers?" (2.3.88-90)
Elizabeth's surprise at Richard's suggestion to go to a museum shows the limits that a racially divided society has put on her own sense of possibility. Even though it's allowed for them to enter the museums, they are the only black people there. Richard hates white people, but he feels he needs to be educated in order to compete in a society where whites rule.