TL;DR: Anyone can go, follow the tape colors, pay the extra money to rent climbing shoes, have fun.
Rock climbing is a pretty fun activity that's generally fun and easy (to get started) for everyone. There are some misconceptions about who can climb, like thinking that your kids are too young, your feet are too big, you're too tall, you're too short, or anything else. From my experience, and from climbing with others, none of these are a blocking problem. They are only one more piece to the puzzle; they require you to figure out how to solve the climbing problems as you are - not everyone climbs the same routes the same way.
If you want to give climbing a shot, it's pretty easy. Most climbing gyms rent equipment like harnesses and shoes, so you don't really need to take anything with you. I recommend wearing longer shorts (like that rule from high school where your shorts must come down to your fingertips with arms straight down at your sides), or even jeans with the ankles rolled up because the harness can sometimes pull at your legs and make it uncomfortable if there's nothing between it and your skin. In my experience, a harness is usually free to use for climbing.
Any shoes will do, but good shoes will make it a much more fun and enjoyable time. Wearing running/tennis shoes or sneakers often make it hard to put all of your weight on a smaller hold, or use friction to hold yourself up. Climbing shoes are (for the most part) designed to give you maximum friction, and a hard, single point of pressure on the ball/toe of your foot. It's worth the extra couple bucks to rent these, if you're going to climb for more than 5-10 minutes.
When you go climbing (if you're climbing indoors, or on an outdoor "plastic" wall), you will probably see lots of tape on the wall, next to each hold. The color of the tape indicates which holds are a part of which route. You can, in theory, just climb using whichever holds you want, but you could also, in theory, make a picture with mixed and matched pieces from multiple jigsaw puzzles - it's just not necessarily the most efficient. This is to say, try your best to use only the holds for the route you've chosen to climb, because there will be a sequence to it that will progress you up the wall the way the route was designed (it's actually very difficult to set good routes).
When you're climbing, you want to make sure you pay attention to what comes next. Don't put your left hand on a hold if the next one after it is way off to the left, for example - because you probably need your left hand for that one. You also want to pay attention to your legs. I put this in bold because I've seen an innumerable amount of beginner climbers get on the wall, only to be struggling to hold themselves up with only their arms. I want to ask them - did you forget that for the last x number of years in your life, you've held yourself up with only your legs? This is very important - your legs are equally, if not more, important than your arms. Your legs give you strength and lift, balance and stability.
At most climbing gyms, when a person climbs, another person belays for the climber. Belaying is holding and controlling the rope for the climber. For top rope climbing (where there is already a rope secured to the top of the wall), the climber ties in, and the belayer pulls out the slack as the climber climbs. When the climber is ready to come down off the wall, the belayer slowly lets out slack and allows the climber to rappel.
Many people just can't grasp the rappelling part, but it's really very easy - just lean back (yes, back) with your hands off the wall, and gently push off with your feet every time you come in toward the wall. You'll be in a sitting position, and it will feel comfortable. If you're rappelling with your hands on the wall and your body in an upright position, you're doing it wrong. Trust me, you're not going to fall, just sit back, and push off with your feet.
What to Climb
Remember all of those taped routes that I told you about? Most gyms rates these routes based on the Yosemite Decimal System (which ironically, is not a decimal number scale). Beginner routes are typically 5.6-5.8, depending on your personal muscle strength and fitness level, and expert routes (at my gym, at least) usually start around 5.11 and go up to 5.14 and even higher. Again, just to clarify, 5.11 is harder than 5.10 is harder than 5.9. I would start on a 5.6 or 5.7, and go from there.
Who to Go With
Go climbing with a couple friends - take one or two more beginners, and one or two climbers with some experience. This helps, because climbers on the ground can help point out the beta (where your hands and feet go, in what order, to complete the route) if you get stuck, but it's good to go with another beginner so you don't get discouraged.
It's also nice to have someone more experienced because they might have their own chalk bag that you can borrow for your climb. Chalk is extremely helpful for keeping your hands dry and smooth, as they will probably sweat up pretty quickly on the wall. Usually you can't rent chalk, but it's not too expensive that you can't borrow some from a friend.
I've seen kids as young as 4 or 5 climb at the rock wall before. Sometimes, special kids' harnesses are available that are full-body, but once the child is big enough for any harness available, it is no less safe for them just because of their age. Staff is usually prepared to help make sure the kids (and everyone, really) are safely tied in, and know the proper procedures.
About my Climbing Experience
Just to be clear about my climbing level, if I break down the spectrum to beginners, intermediates, and experts, then I am an intermediate climber. I've never top rope climbed outdoors, nor have I done lead climbing, but I've climbed quite a bit indoors, and gone bouldering outdoors on multiple occasions. I started on 5.7 and 5.8 routes in January and have been progressed to climbing 5.10 routes for the last couple of weeks.
I'm welcome to answer any questions you have, and if you're interested in trying out climbing, just let me know, and we'll find a gym and have some fun.