Hi, I'd like to know if we can say " why don't we go see a movie" or the correct structure is "why don't we go and see a movie'" or " why don't we go to see a movie". Thanks.
Yep. Aller voir, go see. We do find that complete enough.
Go see is perfectly good English. It has been around for centuries. Both of these are fine:
 Why don't we go see a movie.
 Why don't we go and see a movie.