Well ease her into gaming, try some co-op game first (Fable would be good at first and try to get her to play Gears of war, but spend some time in the arena first, let her carve you a new one so she wont be pissed at you if she dies), DO NOT GO ONLINE, she has to be pretty experienced with offline to start going online.
But try training with her with these games.
RPG: Start with fable and then ease her into Mass Effect then Fallout.
FPS: TF2 (Offline) is good for starters and then try to ease her into playing Call Of Duty: WaW, since its co-op and then let her try Bad Company or something.
TPS: Crash Bandicoot would be good at first then try a Jak and Daxter game and then a Uncharted game (Then GTA and red Dead?).
Puzzle: Portal of course!
Horror: LAST RESORT, when she is experienced enough, you know good reflexs and all.
If you think she is good enough let her Vs you and help her train for the online world and the ease her into online by joining small games and then into the bigger games.
If she does not like these games take her to the store to pick something she likes...and dont complain about the game she picks.
Also I hear Online minecraft is good to chill and relax until...tisssssssss *boom!*
EDIT: Also for fighting...play...ugh, damn, I got no idea mabey...no, crap go no idea for fighting games mabey wait until she is experienced enough to play fighting games since its all about the combos and reflexes and that.