insist on = demand or request in a very firm way.
Demand: (He insists on talking to the manager about the problem, and refuses to deal with any of the employees.)
Request firmly: (He insisted on paying the restaurant bill for all of us even though it was a lot of money.)
persist in = to keep on doing something, despite difficulty or opposition.
(He persists in trying to win her heart, even though she doesn't even know that he's alive.)
1. stick to = to adhere to, physically speaking
(If you put some glue on this paper, it will stick to the wall.)
2. stick to = to continue to do something faithfully
(That cake is very tempting, but I am sticking to my diet because I really need to lose weight.)
3. stick to = to be faithful to an idea or principle
(He doesn't care what people think of him, he's sticking to his opinion on the subject.)
Hope that helps.