What are some example of Atticus's teaching Jem and Scout that skin color doesn't matter in To Kill a Mockingbird?
Atticus' style of teaching his children is not always direct. Instead he likes to teach them through his own actions. Therefore, in the book there is no lengthy discourse that Atticus gives to show the evils of racism, even though he does correct his children. For instance, Atticus does not want his children to use the word "nigger."
“Nothing,” Jem said. “Ask Atticus, he’ll tell you.”
“Do you defend niggers, Atticus?” I asked him that evening.
“Of course I do. Don’t say nigger, Scout. That’s common.”
“‘s what everybody at school says.”
“From now on it’ll be everybody less one—”
The indirect but powerful way in which Atticus teaches his children about race is twofold.
First, Atticus defends Tom Robinson when it was a very unpopular thing to do. Atticus not only put his reputation on the line, but also his personal safety. When a mob came to harm Tom Robinson, they would have also probably harmed Atticus if Scout had not been there.
Second, Atticus employed Calpurnia. In many ways she was like a mother to Scout and Jem. This was a radical thing to do. This is why Alexandra was so shocked that Atticus allowed Calpunia to be like family.