Jen Comas Keck offers some good tips on being more comfortable in the weight room
at the gym. (Targeted at women, but I see lots of newbie men at my gym who look nervous and out of place in the weight room, and I think most of them could use the same tips.)
I will add one caveat to the "People aren't paying attention to what you are doing because they are too busy focusing on the themselves" thing. It's mostly true. But when I'm resting in between sets or walking around the gym, I do look at what other people are doing, and I know that people look at what I'm doing. I don't do it to judge people, although I will confess that when I see someone doing something ineffective or with bad form, I am tempted to giggle or hand them my personal trainer's business card. But for the most part, if I'm thinking anything at all, it's usually, "Go you!" or "I haven't seen that before," or "I am jealous that you can squat more than I can."
Conversely, on the rare occasions I catch someone starting at me, I just assume they're thinking, "Shit, her biceps are bigger than mine." Problem solved.