>>10
If she's kind of geeky, geeky subjects are totally fine but try to limit conversation to geeky things she UNDERSTANDS. Perhaps the most awkward conversations I've had with slightly geeky guys are when they really get deeply into the topic of their fascination (be that anime, gaming, whatever) and it goes over my head. If you want to keep her engaged be sure to talk about things that interest her too. If you want to tell her some neat fact about a subject she's unfamiliar with, be sure to explain things nice and concisely. Things to avoid, though:
-Talking in memes
-Stealing corny lines from anime/manga/video games/etc. Be REAL and sincere.
-Talking about prior relationships. No matter how tomboyish she may be, those're still taboo. My boyfriend didn't get all the details about some of my past relationships until similar situations came up in our own (relationship).

>>7 Seconded; flattery DOES work. You mentioned the girl's tomboyish? Just because she may not be the girliest, most feminine doesn't mean she absolutely does not care about her appearance or how others perceive her. Keep it positive. If you especially like a piece of jewelry or clothing she's wearing, compliment it. Not only will it make her day, there may also be a story behind it that could make interesting fodder for discussion.

Lastly, as far as the money thing goes, I don't think it's necessary to buy her a gift or dinner. However, if you go out for ice cream or coffee or something later, definitely offer to pay. She'll appreciate the gesture even if she doesn't accept the payment.