For movies and television shows, both "watch" and "see" work.
You'll get the point across whichever you choose, but sometimes it sounds better to use one in specific.
When going to the movie theater, you should use "see".
Example: (Let's go see a movie) = Movie theater
(Let's go watch a movie) = Could be anywhere!
If adding "-ing", the meaning changes a bit.
Example: (I'm watching a movie) = I'm CURRENTLY watching a movie.
(I'm seeing a movie) = I'm GOING TO see a movie
For bands and performances, use "see"
Example: (I am going to see a band tonight)
(I'm going to watch a band tonight) = incorrect but gets the point across.
Hope this helps a little!