I try to keep teaching fun and light-hearted. I try not to make it too formal, even for my advanced students. Hopefully students who take my class will gain a greater appreciation of mathematics. Hopefully they will like math more too. This is not my primary goal, but I hope it is often an unintended consequence of attending my classes.
I try to let students do the heavy lifting and figure out the next step in problems themselves. I try to only give hints when necessary during problem solving sessions.
Depending on the class, there could be a mix of problem solving time, guided discussion on one way to solve a problem, or a short lecture to introduction a topic before some practice on the topics. The style and format of each class is different, and can be changed based on current students needs.