You are looking for older games, things like Quake where standing still gets you no-where. These days the prevailing FPS strategy is (With obvious exceptions) to keep people behind a chest high wall in the same spot. A more modern game that you have to run around in, Bioshock. Standing still will get you dead fast, whilst running is the only reasonable thing to do. If that first Big Daddy fight doesn't get her running around like crazy, I don't know what will.
That is what I have taken from your post anyway, that you need movement training for your GF, not accuracy or how to use training.
For 'how to use' training, TF2 is an interesting game. Concept is a shooter, but it has its own twists. It also has numerous tutorials on how to do many things in game, a lot of which I liken to strategies in other games (Hitting your engineers constructs with your wrench repairs them, similar to how using your wrench on a tank that is almost exploded will repair any damage in Battlefield games, albeit slowly [Quickly compared to RL though] and Spies disguising as the enemy I would liken to any game where enemies can be easily mistaken for friends, and you run around a back way to your enemies spawn point and follow them into battle, only to kill them because they thought you were their ally).
Accuracy training I'd recommend something that focuses on long range combat. The Battlefield series has this, especially with some of the mods and larger maps. I remember a remake of Operation Greasy Mullet for a mod that I use that had a 2Km view distance for snipers.
Tactics training I'd probably go Counterstrike for. You very quickly learn that there are certain tactics for each map that work well, and it also brings to light the uses of different types of weapons.
On top of that, just slowly present more and more FPSes to her. She will slowly pick it up. You started off slow too remember, its just a matter of time before she learns what she's doing wrong and improves.
Quite obviously though use Bots for all multiplayer games. Pitting her against players ATM would be disheartening, and she would likely be insulted for 'being a noob'.