As others have said, it's a good idea to find out what kind of movies the girl likes and then take it from there. Stereotypes are meant to be broken, and usually are when you get to know a person. That said, I'm an older chick, and have been trying to get my girlfriends to watch anime. These are some flicks that have impressed them:
Escaflowne: The Movie
Vampire Hunter D: Bloodlust
Millennium Actress
Princess Mononoke (everyone loves this one)
Spirited Away (ditto)
Castle in the Sky
Memories (good because you have three distinctively different shorts)
Trust and Betrayal: Director's Cut
Samurai X: The Movie (you may need to explain some of the background first)
Spriggan
Jin Roh: The Wolf Brigade
Cowboy Bebop (movie or series, but I'd go with the series; everyone loves this too)
As someone else recommended, it's probably best to pick a flick, rather than a series, unless you get stand-alone episodes, as with Bebeop. Also, there's Tokyo Godfathers, which I haven't tried on my friends, but it's pretty appealing across the boards.
I hope this helps. I'm no anime expert, and am learning about it myself, but so far, it seems there is something for everyone, no matter what the person's taste in movies or TV shows.
Escaflowne: The Movie
Vampire Hunter D: Bloodlust
Millennium Actress
Princess Mononoke (everyone loves this one)
Spirited Away (ditto)
Castle in the Sky
Memories (good because you have three distinctively different shorts)
Trust and Betrayal: Director's Cut
Samurai X: The Movie (you may need to explain some of the background first)
Spriggan
Jin Roh: The Wolf Brigade
Cowboy Bebop (movie or series, but I'd go with the series; everyone loves this too)
As someone else recommended, it's probably best to pick a flick, rather than a series, unless you get stand-alone episodes, as with Bebeop. Also, there's Tokyo Godfathers, which I haven't tried on my friends, but it's pretty appealing across the boards.
I hope this helps. I'm no anime expert, and am learning about it myself, but so far, it seems there is something for everyone, no matter what the person's taste in movies or TV shows.