If you don't mind an oldster piping up here. I've had some good first dates. I went for the coffee option, but here is some more input. I assume the eyes meeting across the room has already occurred. And also that this date is for more than just fun. Fun doesn't require a lot of talking and there are plenty of those options.
I like the idea of getting to know someone, without others around and in the light of day with a clear head as a first date. If she isn't impressed, you haven't wasted a lot of time or money. If she doesn't see you as potentially exciting partner, you may have more chances.
Now, to make that good impression, choose the atmosphere. Lunch might even be better than coffee. As long as it isn't burger, pizza, KFC. Casual, not stuffy. Go for a sandwich/salad/soup place and make it leisurely. Avoid the topics of government, politics, and only skim through the family stuff. Maybe talk about the food. Or travel or animals or nature.