Just because a girl says, "Pick whatever movies you want," she doesn't actually mean that. What she's really saying is, "Pick the right movies and I'll be impressed that you were thinking about me and you listen." That's the secret.
Here's what you're going to rent - films that will impress your girl and not make you feel like you've sold out. Watch in this order, and I'll give you one reason why it will blow your mind and make you seem like the best boyfriend ever.
1)The Graduate
For guys: What man hasn't been frustrated by not knowing what to do in life, or attracted to an older woman? Smart, witty dialogue.
For girls: Character makes the ultimate romantic sacrifice at the end.
2)The Insider
For guys: Holy crap is Pacino good in this.
For girls: Like the middle act of a play, save the darker one for second. She might get bored, or she might love the story and amazing script, depending on her personality. Doesn't matter though - you're going to end on a good note.
3)Chocolat
For guys: Juliette Binoche is hot, and the story is whymsical and engaging.
For girls: They seem to dig Depp for whatever reason.
The winter months are great for movies like this. Stay away from explosions, foul language and 4 hour epics, and you'll be thanking me later.
Cheers.