There are two schools of thought nowadays when it comes to paying for a date:
1. Whoever invited the other on the date pays. This is based on standards in Europe where whenever you invite someone to do something, you assume the responsibility to pay for them. If you say, "No, I want to pay for myself," they'll say, "No, I'm inviting you." If you're ever in Europe and someone uses the word "invite," know that they're saying they're going to pick up the bill. This is something that in the last decade has hopped across the pond and become prevalent in America, too, especially when it comes to dating.
2. You each pay for yourselves. In these times of sexual equality, what's called "going Dutch" has become standard fare, too. In particular with heterosexual dating, a woman often will insist on paying because she wants to avoid the strings that lots of times come attached when a guy pays, a long history of guys feeling like they're owed physical contact or intimacy after having paid for a date. In fact, it's become sexist for a man to presume to pay for a woman, like she can't take care of herself. The way out of that is for the man to jump to number one and say, "I invited you, so I really insist on paying, but if you've had a good time and want to ask me to go out with you on another date, which I'd really like, then you'll be inviting me and so how about you pay for that one?" That also has the added benefit of sharing the responsibility of planning the date and dates not always being what the guy plans, because if she's taking you, then the guy will be going along with her plans.