It's funny, when I was in high school and college I felt exactly the same way. I was incredibly shy and self-conscious and was always afraid people would watch me or talk about me. I hardly left my dorm room, much less worked out in public places.
It's been 20 years since college, and when I look back I wish I had known one important thing: people just don't care

Now that sounds a bit harsh, but it's actually comforting when you think about it.
For example, when we do or say something embarrassing, we tend to think it's the end of the world and that everyone's watching us and talking about us. The truth is, maybe they notice for a little while, but they forget about it in minutes, even if we take hours or days to forget about it.
So my advice to you is to not do what I did when I was young. Don't stay locked up for fear that people will talk about you. Force yourself out of your comfort zone and get out there. You are doing a great thing--something a lot of people don't do--by committing to go to a gym and get yourself in shape. You have as much right to be there (in fact, more so) than anyone else...your money (whether you pay it to the gym directly or through your tuition) is just as good as anyone else's.
Yes, there's a tiny change there be some guys who never grew up who might immaturely judge you. But you know what? Who cares about them? They're going to be jerks to anyone whether they're in a gym or outside of one. And somewhere down the road, they're going to pay the price for being a jerk. In the meantime, you are going to get yourself in amazing shape, you'll look great, you'll feel great, and you'll live a long, healthy life. Don't let anyone steal that away from you.
Good luck!