I'm a hearing guy dating a deaf girl, and the biggest advice I have for you is to
have patience, both of you. I took a college ASL class last fall, and I'm improving vastly, but I still have a
long way to go. My girlfriend is very patient with me and my slow receptive skills (learning to read sign is a lot more work than learning the signs yourself); we both know there will be miscommunications, and take the time to make sure we are understanding each other properly. It takes some work, but when you really care for the person, you don't mind it.
As far as learning sign language, you can usually find a class at most colleges and universities; see if you can audit the course. Also, Deaf community centers and some churches offer sign language classes. But, to get you started right away, visit
this website to see signs in action.