I have pretty limited dating experience. However:
If the first date is for coffee or drinks, I show up a few minutes early and buy my own. If things are going well and he offers a second round, I accept.
If the date is for lunch or dinner, I make sure it is a restaurant I can afford (just in case) but honestly expect him to offer. Second date you should offer. Once you start dating someone regularly you should talk about expectations, but often at that point you take turns picking up the tab.

Interesting note: Studies have been done, and if "the check dance" is done at the end of the meal, the guy is 80 times more likely to consider a second date. "The check dance" is when the check comes, both people are expected to reach for it, the woman says something like "would you like me to get this" and the man says "No, I'll get it" then the woman demurs politely and smiles. According to the studies men say that if the woman does not reach for the check they feel used, and if a woman insists on paying than they feel like she is not interested.