I never really had to introduce my girlfriend to gaming, she was already a gamer when we met. That said, she has branched out and tried a lot of games she normally wouldn't have tried on her own. When we met, she was more of a survival horror, RPG, and the occasional RTS girl. She didn't pick up shooters too much until she met me. I helped expand her genre tastes: She plays many more shooters and RTS games now. Likewise, I'm more willing to try games I normally wouldn't have tried because of her.
But...We often run into problems when trying to play together online: The maturity level of the average gamer. Unless we are playing a private match only with our friends, she won't use the mic or give any hint that she is a girl. The average gamer on Live, upon hearing a girl, either begins spamming her with friend requests and vulgar pick up lines, or starts going into some sexist rant. It isn't too bad on the PC and PS3, but on the 360...Dear God. It makes it hard for us to really enjoy our time together when playing over Live because it happens in EVERY SINGLE MATCH after she uses her mic. Constantly pressing the mute button gets really irritating.