Insist on means that you are firm in your wish to do something.
I insist on helping my mother with the shopping even though she says I don't have to.
Persist in means to continually do something over and over, usually a negative thing.
He persists in shouting, even though we told him to be quiet several times.
Stick to is an informal phrase meaning to stay with one option.
The blue shirt doesn't suit you; I think you should stick to wearing black shirts.