I second (third? Fourth?) The recommendation for Community. Probably my favorite sitcom on television right now. Parks and Recreation is also great once you make it through the first season (six episodes). 30 Rock is funny, though it takes about half a season for it to hit its stride. How I Met Your Mother is funny and not too taxing, as others have said. If you haven't watched it yet, ReZ is right about Arrested Development being awesome. It will require your full attention, but it's a great show to watch with/at the same time as someone else.
If you want something that mixes comedy and drama, Ashley is right about Chuck being fun, even if it's a bit predictable at times. I also really liked Terriers, though sadly it was canceled after only thirteen episodes. Great show while it lasted, though. It reminded me a bit of Veronica Mars, which is also really good if you haven't watched it yet.
Breaking Bad, Mad Men and The Wire are all fantastic shows, but Ashley is right about them requiring your full attention. I think the first two shouldn't be too taxing, but The Wire is very emotionally draining, to the point where it's almost painful to watch at times. But it's also utterly brilliant, so if you think your girlfriend would be into it, then definitely watch it.
There's also Dexter, Six Feet Under, The Sopranos, The Walking Dead, Boardwalk Empire... I know you said she wasn't into science fiction, but if she's open to it, you should at least try to get her to watch the miniseries for Battlestar Galactica. She might still like that (Caprica is great too, but also science fiction).
Basically I recommend every TV show ever.