Hi Jessica,
Same sorts of ideas as Janet and Jenny. Coffee and a chat. Dinner, lunch, brunch. Depends what you want to achieve, so to sound clinical. Women generally like to talk and discuss their relationships, circumstances, emotions and thoughts. And I have very naturally fallen into this. Guys generally don't. One reason my therapist decided I wasn't a guy. So a place to chat is good.
Going to a movie is entertainment, again fine. But why? To go out with someone? Prove you can stand looking at a chick flick (I can't, many if not most of my GG friends can't either, it may be an age issue).
So you can demonstrate that you pass in public, or that you don't, and want her help to do so?
So, I hope I'm not being negative. You are in a great situation of going out with a female friend. BTW I do often go out with my sister in law. We go to restaurants, theater etc. Her husband knows me totally, we go to restaurants on the night he goes to his quiz night, and to theater that he doesn't like. We go as a threesome to movies we all like.
So decide what you want to achieve.
I would suggest going shopping together, although great fun, will be a little down the line when you know each other in your female persona. Then you can really chat about clothes and styles that suit and you both like, all the great stuff.
Hugs
Cindy