Dr. Strangelove or: How I Learned to Stop Worrying and Love the Bomb is my favorite movie. It's got great characters, including three played by the same actor, is well written with some very quotable lines, and the story is both absurd and believable (even now).
I'll be the first to admit that it's not for everybody, but if you're asking for movies to watch it's probably for you. With movies I always recommend going in knowing nothing.
I could quote the whole thing probably, but I'll spare you and just say that the way the Russian ambassador says "fresh fish" is weirdly seared into my brain.
It's incredibly weird that roughly the same story was adapted, the same year, into two movies, one serious and one comedy, and... they're both masterpieces.