Try to throw in a complement or two as you ask her out. Something about how you think she's nice and would she like to go on a date with you.
If she says no, gently follow-up with something like "maybe another time?" and see what she says. If you at least get a maybe, then just ask her again in another couple of days or try to get a general timeframe from her (next week, tomorrow, after some event, etc).
As for what to do, public events are good. Local fairs, etc. Maybe a museum, the zoo, aquarium, etc. Find out what other people do on dates in your area, but don't get caught up doing romantic stuff on the first few dates. Just try to find something to do that will not put her on the defensive wondering when you're going to "make your move". That means if you go to a movie, sit in the middle or up front, not in the back, so you're not tempted to do something stupid like grab a feel or something and she's not worried that you'll do something stupid like that. Holding her hand in the scary parts - that's cool especially if you see her hands jumping, but don't push it.
Anyhow, back to just asking, I wouldn't recommend being too cute about it. Just try to work in a complement and ask straight out. You should have in mind what you want to do so if she says yes, you can ask her about your date plan. "yes". "great. I thought it might be fun to go to the roller rink" "No, I lost my foot to a land mine". "oh, I'm sorry. How about go-karts and miniature golfing?" "Great" "Can I pick you up at 5? We can be home by 10 if you have a curfew" "yea sure" "See you then!"
When you pick her up, have a single flower to give her so she can hold and fiddle around with it in case you have any "uncomfortable silences" on the way to whatever it is you're going to do.
Just some ideas.
