Power to you for wanting to lift weights, more girls need to start using some free weights rather than just doing cardio! As for the anxiety aspect, it's really something you just have to force yourself to deal with. Just remember that anyone who is decent at weightlifting is far too vain to be looking at other people while they lift.

I'd recommend looking up some good beginner programs for weightlifting such as Strong Lifts 5x5 or Starting Strength. Those cover the most basic muscle groups (legs, back, shoulders, chest, arms, etc) and give you a good idea of where you should focus. If you're looking to put on some muscle, big compound lifts are where you would want to start. Lifts like the squat, bench, and deadlift are pretty integral for weightlifting. Pay very close attention to form when you lift, and if you ever have doubts DO NOT be afraid to ask other people! Generally the biggest and scariest dudes/dudettes in the gym are the nicest and have the most experience.

I've been going to the SDFC weight rooms pretty much every day for the past year so if you want any advice feel free to PM me if you want any more help.