The first anime movie I ever bought was my beloved Vampire Hunter D (and the sequel, Vampire Hunter D: Bloodlust). It was also the first book series and manga I purchased on my Kindle Fire. I'm 33 years old, yet it's hard to believe that the first Vampire Hunter D movie is 30 years old today. It was the first film my husband and I watched together (bonding over similar tastes in film and games in the hours before a session of Vampires: the Masquerade). Man, even THAT was 13 years ago...woo, feeling the years... But it's a great introduction into anime! The story takes place in a post-apocalyptic world, where the rules of survival are a little different than we'd imagine. Vampires (mutant human creations) roam the wasteland and prey on unwitting humans. D is a wandering mercenary and a dhampir (half-human, half-vampire). He has a particular hatred for his kind, hunting them down one by one. The film has a fascinating mythos and stands the test of time fairly well, though the graphics are a bit dated. And Left Hand, his demonic symbiote, is pretty much the life of the party!
If you haven't seen Vampire Hunter D, I totally recommend it (and the sequel). There have been rumors of a new anime series in the works, so that might be worth a look when it comes to fruition.