"I'll meet you at the movie theater at 8 pm" is good. "I'll meet you at the movies at 8 pm." is also good. "I'll meet you at the movie." would also be good. "Let's go to the movies." also works. If I were asking someone if they wanted to go, I would say "Wanna see a movie?" or "Wanna go to the movies?" ("Wanna" being the commonly-said quick form of "want to")

I would only say: I'll meet you at the movie theater. The movie theater is a place, a location. "The movies" is not a place or location. It is a general reference to a concept.

Probably the best translation in AmE for "the cinema" is "the movie theater" as you say, or simply "the theater". (Note that in BrE, "theatre" would never be understood as a venue for films, but just as a place for live performances. It is used that way too in AmE, but the context would have to inform you which meaning was intended.) The word "cinema" is used sometimes, but more to mean the activity of film-making than a venue. You could indeed say "go to the movies", but that would have the sense of "go to watch a film" rather than going to a particular place.

Okay, so in the United States, we say some things differently. In the U.S, we say "movie", and these are "films" (U.K. people and some producers use 'film') that are played at a "movie theater" (or, "the movies"). It's confusing if you've learned English (U.K.) because you hear "want to go see a film at the cinema?" A "theater" in the U.K. is a place where plays and other live performances are done, where as in the U.S., a theater is where movies are played. To answer your question: it's really a matter of who you talk to. I'm an American born and raised, but I say "film at a cinema", and I might even slip in the accidental "chips" instead of "fries" or the "telly" instead of "TV". Point is, it's a matter of where you go. Some people say movies, others say movie theater, some say cinema. At the end of the day, in the US, they all mean the same thing.