Don't pretend you are making a joke. She's not dense. Just look at her and extend your hand for her to take.
Don't do that unless if she's already been flirting with you.
Go to a non-chain restaurant. Not fast food, not Applebee's. For example, a good diner or a pizza place with a nice atmosphere. Pay for it yourself, and if she argues make sure to argue back. Don't be a wimp.
Say "thank you." If you're going for damage control because it's clear she only thinks of you as a friend, just tell her you had a wonderful evening. If you think she's into you but she doesn't make the first move, ask if you can kiss her (it's romantic). Don't force yourself onto her-- it's creepy, because it's apparent you're not that kind of guy. Don't hug her, that's totally awkward.