Yeah, climbing gyms are great places to start because the pretty discrete nature of climbs makes learning technique easier, I think. Outside it might not be as clear exactly where to put your hands and feet to make a particular kind of move, but inside, hey, there are colored holds bolted to a relatively flat wall, so there's less confusion.
Another great thing about climbing that is helpful when you're getting started is that everywhere I've ever climbed, the community is almost entirely made up of friendly, helpful people. Climbers tend not to be as self-conscious as people working out in more typical gyms, in my experience, because the focus is on improving climbing skill, not explicitly on sculpting the body. And by and large, they are all happy to offer tips (known as "beta" -- like, you might ask, "Anyone have any good beta for that tricky mantle move halfway up?") and help out.
I primarily boulder; bouldering is climbing low to the ground -- shorter, more technical routes, but you don't need as much safety equipment, harness & rope and whatnot. But if you find you like climbing longer distances, you need the harness and rope and whatnot, which they'll have for you at gyms, but if you are climbing outside, it becomes a bit of an investment, potentially.
On the upside, you *can't* do roped climbing without a partner, since you need someone on the other end of the rope, so it makes for a great social activity and if you schedule it regularly with another friend who wants to climb, you really can't bail on each other like you could at the gym. If one of you doesn't go, you know you're spoiling the day for the other guy too. Cuts both ways, though; that difficulty of scheduling is a big reason I don't climb outside more often.
In any case, I highly, highly recommend it as a sport. Bouldering is naturally like HIIT (high-intensity interval training) since you work on a route and get pumped for a minute or two, then take 3-5 minutes to rest between the next one. So it's great for weight loss, I've found, and builds really nice upper body strength. Well, okay, your forearms will get somewhat disproportionately large, but not to Popeye dimensions, and on the upside, you'll be forever able to assist & amaze your friends and family by opening even the most tightly-sealed screw-top jars.