Both phrases are completely correct in North America. Here are some related examples from Practical English Usage 3rd edition by Michael Swan, p45. section 53 subsection 2 - come/go and ... excerpt "Come and ..., go and ..., run and ..., hurry up and ..., stay and ... are often used informally." "Come and have a drink" section 53 subsection 3 - American English "In informal American English, 'and' is sometimes dropped after base forms 'go' and 'come.' " "Let's go see if Anne's home."

"Watch" is correct and more appropriate, however certain people (Americans especially) use "see" too. The reason you would use "watch" is because you will be observing over a period of time (the duration of the film that is), as opposed to merely looking at something or visiting someone/something.

Hi Kai! First of all I need to say that the way you wrote both sentences are incorrect because you should have written "go to". In my opinion I would say "Let's go to see a movie" when you are referring to go to a place (like the cinema) while at the cinema you are "watching" a movie. I hope that this helps you!