First, I would use console games, being in front of a tv together feels like a social activity much more so than sitting behind two different computers. Get games of every genre and type, (party games, fps, strategy etc). I would start with the party games, as they often include many mini games. The idea is to find something she likes/enjoys, and then to explore the entire genre with her, and then pick another. Games that she has a natural aptitude for and that you generally suck at will be great. Also lots of games will balance the playing field out if you are doing poorly (Mario Kart!).
And remember to keep it fun. If she's bored with the game don't insist you keep playing, just find another, like you said you have lots of experience, and just like you it will take her some time to refine her tastes!
Also for the console experience, the PS3 is a great choice as online purchased games can be installed on more than one PS3. So you could have a two tv two ps3 setup in the living room (head to head FPS games, tetris, the new god of war).