If your girlfriend likes Miyazaki and you personally are trying to find an anime you'll survive through, you've got to watch Dennou Coil. Dennou Coil is absolutely amazing!!! Think light, refreshing Miyazaki with a techie twist. It's simple, romantic, sad at points, funny at others, suspenseful, exciting, mysterious, and absolutely breathtaking where the art and animation is concerned. It's 26 episodes, which is what I think is a decent length for an anime series, and I enjoyed every second of it. Watch it now!
I'm going to split this up into categories. Pay attention now. 8D
If your girl is feeling like something HAPPY or CUTE or FUNNY: Most of Ouran is great, if she can understand the reverse harem genre. Try and stay away from the last few episodes, which are pretty depressing. Lucky Star is pretty good for the wtflol factor. Eyeshield 21 is pretty good if you're desperate for testosterone -- it's hilarious all around, you get American football, she gets hot guys to ogle. Wait, sorry, that came out wrong. XD The only other anime I can think of to suggest here is Yakitate Japan!, but it's a little pun-heavy and I'm not sure if we English natives can appreciate it fully.
If your girl wants some DRAMA or TRAGEDY: Voices of a Distant Star would be number one on my list -- it made me cry in a good way. I saw someone mention Air, so if the art style doesn't put you or your girlfriend off, go for it; your girl will be hooked by episode 2. If you get to the end, the story gets really abstract, so you two might want to pause and let it soak for a while. Go for Kino no Tabi if she's feeling like an adventure, or, if you're really brave (or have really strange taste... like me...), give Mononoke or Mushishi a test drive. Finally, Gunslinger Girl is wicked sweet and everyone should watch it.
If your girl wants ROMANCE: Again, Fruits Basket would be GREAT, if you care to pick and choose episodes (some are really, really depressing). It's wonderfully romantic.
If your girl wants ACTION: Along with Cowboy Bebop, I would suggest Baccano!. It's smart, classy, lots of fun, and best of all, it has hot guys in it. Wait, sorry, that came out wrong.

It's about the Mafia in Depression Era America. The story is well-planned, twisty as all get-out, and satisfying. Check if your girl can handle gore and violence, though (here's a little preview -- tell her a small boy's head gets completely blown off by a gun at point blank range within the first ten minutes). Otherwise, Tengen Toppa Gurren Lagann is a solid piece of eyecandy.
If your girl PLAIN DOESN'T KNOW or CARE: I would try Clannad. It has a little bit of everything: action, romance, comedy, drama, etc, etc, (with lots of JRPG nods if you or your girlfriend's into them), plus the story just gets plain addicting after a while. If you stick with it after episode 4, bring some tissues.