I've just started dating this girl
Ice skating sounds good. You must be in a chilly climate. I love walking and hiking. A good movie is fun, if you like discussing motivation and such afterward.
Wine tasting, I love it! Doesn't have to be great stuff. A cooking demo.
How about cooking/baking together...something simple or if the two of you are adventurous or at least have good sense of humor, maybe something new for both of you.
It's really nice of you to want to do something new and different with your girl!