The majority of the advice given so far will help you out. Here's a little more:
Do not offer her a drink until you've been socializing with her for half an hour or so. If you offer her a drink in the first 10 minutes or even worse, you buy it and have it sent across the bar to her before you even get near her, you just wasted your money unless she is on a mission to get laid that night. And even if she is, she probably won't make it quite that easy for you. I can only imagine how much money some men have spent in their 20+ years of buying a woman a drink within the first few minutes of hanging out with them only to leave the bar with their drinking buddy. Buying a drink for a woman right off the bat is the most obvious mistake men make in bars, yet several of them continue to do it well into their freaking 40s. It basically yells to the woman, "I'm trying to get you drunk, so I can bash you out!"