Yes, it is fine, and natural, to say 'see' for films, especially if you go to the cinema to see them. You can also say 'see' for films and TV programmes which you see at home.
NB I'm obviously using British English vocabulary and spelling here, but the verb use of 'see' is the same, regardless of whether you use US or GB English.
And yes, you do need to put 'a' before 'movie' and 'TV program' in these sentences. These are singular countable nouns, so you have to use an article.