If it's your first time, you'll probably work your upper body a lot (unless you already participate in a sport where you really use your shoulders/back/core). The day after, my shoulders, core, and forearms always seem to feel it the most. A lot of people think climbing is just your arms, but you'll use your legs a lot also. It's a lot easier to just step up and use your thighs to lift yourself than it is to try to pull up your body with just your arms. I think it's a pretty good all around work-out and it's lots of fun also.
Most gyms will have shoes you can rent. But bring socks if you don't really want to put your bare feet into something lots of other people have worn. You don't need gloves. Someone on staff can help you fit your shoes.
You may also need to rent a harness depending on what you decide to do. If your gym has a bouldering area, you won't need a harness as there aren't any ropes involved. However, if you decide to do some rope stuff, you'll need a harness. Again, someone can help fit you with one. And they'll most likely make you take a belay course just so you know how to tie your knots and do all these safety checks and stuff. I'm partial to bouldering since it's less "equipment" needed, but I prefer bouldering over rope stuff anyway (indoor or outdoor).
If it's your first time, take a friend if you can. It's definately more fun with a friend ... and you can belay for each other. Although some gyms these days have these self-belay devices.
If you decide to go alone, gym people are usually pretty friendly. Just go up to a group and introduce yourself.
There will most likely be routes already set up and they're usually marked by degree of difficulty using some sort of color system (my experience in all the gyms I've been to). However, you don't necessarily need to stick to those. Just getting up and doing whatever you want using whatever holds you want is perfectly fine (and still fun!).
Have fun!