My guy and I game together. I have always like video games, but am also rather cheap, so he is more experienced/has played longer. Luckily we seem to have the same interests, or at least enough to give almost any game a shot.
Currently our set up is a large desk with both our computers set up side-by-side and a couch in front of the desk for our chairs. It works great. If we are playing something together we can see eachother's screens and easily reach over and click a button or two as needed. Sometimes we play the same game, sometimes we do our own thing, but the side by side set up works rather well for us. For the OP it would actually work well. You could set your GF up on her system playing Mass Effect while you do your own thing. Since you have played it you can help as needed, she can get practice with a shooter, find what works for her, and if you start getting irritated you can avert your eyes back to your own screen.
He really does not have the patience for games like WoW that take planning or coordination. He prefers to actually get in the game, play, then log off with nothing else. I actually enjoy other aspect of WoW, I like achievements, I don't even mind farming, so I guess that part is different, even if he was the one that got me started in WoW.
In a lot of ways we like very similar games just play them differently. With an RPG he will dive in and just play, does not care much if stuff is missed, and if he gets lost somewhere that is a higher level, he is even happier. Me, I like to read about secrets and such and try to get a more complete game. I hate getting into a game then reading that there is some really cool thing i could have done/been doing but did not know, so i just read up first. I also tend to enjoy leveling more than he does. I would rather run around and be overpowered on the boss, he would rather just try the boss and not care if it is a long fight, me I like to be overpowered. I also read up on it because if I don't I end up spending hours running around clicking on everything I can find since I just don't trust games to not have secrets, and left to my own mind there is no end to the ideas of pixel looks off, or what strange combination of actions at a certain pixel might yield some other result. Often what happens is he gets me to play a game he likes, I end up liking it, reading about it, and then start telling him stuff that he did not know about.
The other thing I have noticed is out attitudes to mana. He frequently ends up favoring mage type characters as magic is powerful in games. I tend to avoid magic characters as I tend to get too preoccupied with mana conservation to use it often enough to be useful - I hate getting low on mana, so I tend to use it far too sparingly. Again I am just a bit more conservative and he is more the "who cares let's just see what happens" type.