I think the first game my wife and I played together was Smash Bros. Melee at a friends house. It's a good choice of a starting game because it's easier for button mashers to keep up. We've been married five years now, so we've played a lot of other games since. The first game we beat together was Guantlet Dark Legacy (it's still saved on my memory card.) When MS gave away Undertow not so long ago we've decided we like that. We've played the X-Men Legends games together too plus Mario Kart on the GCN and the DS. We were big into Donkey Konga and Guitar Hero 2 for a while too (in fact she busted out the bongos last night while I fed our son.) I guess most recently she was a huge help in Super Mario Galaxy. Wii Sports and Super Mario 3 on the VC are ones we enjoy playing too.
My wife only started playing video games when our son became old enough to get interested in them. So our first joint game with all three of us together was either Godzilla: Destroy All Monsters Melee or Billy Hatcher and the Giant Egg, both for the Gamecube...
My current gal doesn't play video games. I had one previous one that liked some retro games from nes/snes when she was young. But if I could get my current girlfriend to play it would be basicly anything coop. Probaby a Wii title or Rockband/Guitar hero would be best.
My ex-wife and I use to play Super Puzzle Fighter on the PS1. We had a lot of fun with it and I still enjoy the game despite the marriage not working out. Off the top of my head, I can't think of a game I'd want to play if I decided to date again.
Skip-Bo on the DS. It was a battle, but I've finally gotton my wife playing games. She is more of a casual gamer so finding games that we can play together and both enjoy is the issue.
I have found that many girlfriends I've had, and plain female friends, throughly enjoy a good Mario Kart game, although one was partial to Diddy Kong Racing.
I would like to play a Co-Op RPG(and I mean real Co-Op, not just "bucket carrier"). Mario Party or Mario Kart would be awesome too, especially the 64 versions, they never get old.
I have to harken back to the days with my Ex-Wife.. we played Ultima Online... she was really into the game, and it was cool, to find something we could do together...
I still remeber "guarding" her as she would spend hours trying to tame horses.....