Never take a women to lunch or dinner on the first date. You are stuck with that person for the duration of the lunch , and no matter how much you might want to be somewhere else you are stuck there. It ain't fair for either of you :P Try something more casual than a lunch. A coffee, a drink, the walk you mentioned , whatever something which doesn't commit you to any large extent of time. See how that works out and take it from there.Can I ask what age you both are?
A date doesn't have to be a movie, you could go for lunch somewhere nice or even a walk in a park.