As an NES, but not a teacher:
a) I would rather have said: "I am going out to watch a movie." "To go for something" tends to suggest something different as in "Go for it!" which means put all your effort into achieving it or "Do it!"
b) I am going to the cinema to watch a movie.
A theatre is where you would normally watch "live" entertainment such as a play, whereas movies are normally screened in cinemas.
Hope this helps