To the OP, you are and will continue to receive some of the best advice on here from all these experienced members. I think the best thing is that everyone on here is very helpful and give very diverse views on what could work for a new female shooter.
Me, I'm Chinese, so I can understand some of the cultural differences you may be facing with your new GF. As everyone has mentioned, you can't push her and still expect her to have fun. Most of us chicks tend to move at our own speeds according to how we feel on a particular day.
You know, some days we want to bake you a cake and on another, we just have to kill something!
Like one post said, watch out when she gets hooked! You'll have to constantly update your shopping list! (As if that's a bad thing, right?)
The 10/22 is a great starter rifle, but as a pistol shooter, my tastes tend to go towards handguns. You never know -- she may like all this so much that she'll become Double Distinguished!
Best of luck to the two of you... it's a cool way to spend time together.