Some important things to know when you're climbing in a gym are some of the safety guidelines and a lot of the etiquette that goes into being able to really be a part of the climbing community.
Things to consider safety-wise are, you don't want to be running around in the gym. Really try to keep that pace down. There are lots of ropes hanging, too. You don't want to be swinging on them. Leave the top ropes hanging where they are. Don't grab them, don't push them around. There's often times belay devices clipped to those ropes, and one of those can swing out and hit somebody, so don't touch the ropes.
As far as general voice volume is concerned, too, try not to yell. There's really no need to scream in a gym. It's already pretty loud, with music playing, with everyone talking, with folks just mulling about. Really importantly, while folks are climbing, top-roping, and leading, the climber and the belayer need to be communicating, so if you're yelling about something on the side, it can make it really difficult for those climbers to hear each other, which is a really important part of the process.
As far as other safety guidelines, you want to make sure that you're aware of other hazards in the gym when you walk in. Try not to walk underneath other folks that are climbing. Try to walk around that pair. If you see a rope hanging, just don't pass underneath it.
As far as some of the etiquette is concerned, a lot of those types of etiquette issues are summed up with just common courtesy. Thinking that, if there's a lot of folks waiting to do the same climb, just try to observe what that natural rotation might be. If someone's waiting, then feel free to get in line. Ask them if they're waiting.
Maybe you could find another route that's open to do a couple laps to warm up on before coming back to it. If someone's climbing, and you put a lot of pressure on them to rush so you can get on it, it kind of makes it uncomfortable for the climber, and when there's lots of terrain to choose from, really try to avoid putting any undue pressure on someone to rush through a route so you can get your turn.
Similarly, with bouldering, you'll notice that lots of times when folks are climbing, other people are sitting around the mats, observing, should be spotting, things like that, so once again with that rotation, just kind of take notice. Usually folks alternate through in a logical order. So after you've had your turn, maybe you've had an attempt, go ahead and step back, let someone else give it a try. Try not to just hog that whole space and continue trying that one move over and over again until you get it. Let other folks hop in.
Also, don't hesitate to ask. If someone else is trying a boulder problem, and they're making it look really fun and you want to try it to, just ask them. Ask if they mind if you hop in, if they wanted to work that boulder problem together.
Climbing is a very communal sport, bouldering in particular, so don't hesitate. Interact with the other climbers. Communicate. It's supposed to be social. Just get involved with other folks at the gym.
Another really important issue is when you see somebody bouldering, you should always be spotting. If you're sitting around and you see somebody climbing that doesn't have a spotter, please stand up, help that person out, provide a spot for them. It's really important, it's really good to see, it makes it safer for everyone. If you spot other folks while they're climbing, they should extend the same courtesy to you.
Don't hesitate to ask someone to give you a spot. If there's a particular climb you want to do, and you don't see someone around, go find someone. You could go to the front desk and ask if maybe there's a staff member that's available. but having a spotter is critical, and it's part of the general safety of rock-climbing.
Another etiquette point might also be that you'll see folks walking around the gym with brushes, cleaning some of the climbing holds off. Something climbers do to remove some of the grit, some of the sweat, and the chalk that just builds up on those holds. If you see someone brushing a hold off, they're probably brushing that hold because they want to give that boulder problem or that route a really serious attempt.
If someone brushes a hold for you, same principle. Brush it for them, or even ask someone if you can brush the hold for them if you see that they're trying really hard, and then maybe they'll do the same thing for you. It's just part of the communication, something to be open about. If you see somebody brushing, take part, see if you can help, see if they can brush for you, you can brush for them. It's a nice trade.
That's some basic safety guidelines, a couple little points of etiquette, some things to get you going here in the gym, and I hope you guys enjoy it.