To me, the most important lesson the kids learn from Calpurnia is that they need to treat people well no matter who the people are. This is not something that they only learn from her, but it is the most important lesson that they learn in the book.
I think that they learn this (or at least Scout does) most effectively when they bring Walter Cunningham home to eat on the first day of school. On that day, Calpurnia quickly corrects Scout when she seems to think she is better than Walter.
By this and by her own dignity even though she is herself low on the social scale, the kids learn that all people deserve to be respected.
One of the major things that both children learn from Calpurnia is the difference between their world and that of the black citizens of the town. By bringing the children with her, particularly to church, they get to feel some of what her world is like and begin to understand that it is very different. In a particularly valuable moment, they also get to see what it is like to be the object of discrimination.
Another incident that she uses (though not necessarily directly) to teach the kids is when she hears about the rabid dog and she immediately calls Atticus. After the kids had been complaining about how Atticus wasn't trying hard enough to be famous or that he wasn't as tough as some, he shows up and puts down the dog with one beautiful shot and they learn of his reputation for being a good shot and also see his coolness in a pressure filled situation. Calpurnia seems to direct some of these opportunities even if she isn't in direct control of them.
Calpurnia sets a good example for the children. From her they gain nurturing and discipline. Calpurnia focuses on teaching Scout manners and appropriate social skills. Scout had a tendency to blurt out what she thinks. Calpurnia and Atticus compliment each other as people raising the children. It is not really Calpurnia’s position but she has cared for them since they were little. She is the mother figure in their lives.
Calpurnia teaches the children tolerance and the differences between the lives of white people and black people. When she takes the children to her church she is protective of them and shows them how different her speech is among her black peers.