I want to say "I want to watch a movie with you"..
Can I say this?
あなたと映画を見るつもりです。I plan to see a movie with you.
と is the right particle to use with あなた if you want to mean "with them", however, つもり tends to mean "I plan to" , and not "I want to".
I think this is better:
一緒に映画を見たいのです。I want to see a movie with you (together).
The の particle I put in for "explanation" towards the listener. You are trying to explain to them you want to see a movie with them, correct? The particle isn't necessary, it's just my preference.
Note: The sentence I've provided you might not be used if you are asking a good friend (it's kind of formal). In that case you could say:
一緒に映画を見たい！I want to see a movie with you!