K.K.
J. J.
I was a competive gymnast for 15 years. Each "gym" is different so it will really depend on what you are looking for in that some gyms are what I call "hardcore real" they are training young girls to compete in the sport. Some are are just "play gyms" They have mats and bars, but they instead of teaching the art of the sport in terms of competing, they focus mainly on gross motor skills and just a good time for the children. And, you will probably have a lot of luck in that most gyms probably do both. They will start out as gross motor skill development and seem like play. But, if a coach sees potential, they will talk to the parent about the child. Prices can be very minimal to very expensive (100's a month). I would say that if the price range is in the higher end, that is probably going to be more of your "hardcore" type gym. Also, you might be able to find a "mommy and me" gym. These are great for new comers because instead of watching form the seats, you will be out there with your child helping her learn to do the rolls and cartwheels. The instructor will in tell you what to do to help your child learn the trick.
When I was coaching 3 to 4 year olds, these are some of the things we worked on:
Vault: Learning how to run down the mat and hit the spring board with two feet. Working on using the spring board to get onto the vault and sort of squating position then jumping off the other side and landing. As we progressed, we learned to do this in one continious motion.
Bars: Strenght. Pulling up chin up over bar. Learning how to swing back and forth adjusting your hands as you go. Learning how to get onto the bar doing a "back pull over" and then holding your body up.
Beam: How to get on the beam properly. How to walk frontwards and backwards on the beam on toes. Scales, turning, frontrolls, backrolls, how to dismount by jumping off the end.
Floor: Frontrolls (how to tuck your head under so you don't flop over), backrolls, cartwheels, handstands, leaps, etc.
For a 3 year old, I wold say the only "gear" would be a leotard and some shorts (bike shorts, gym shorts type). If your child has long hair, pulled up out of the face. Low ponytails work best because until you learn how to adjust, learning a front roll on the balance beam with a ponytail can be a challenge.
While I trained in the competitive area, I worked in gym that was more on the "fun" side. Some girls competed, but most of the classes were for fun learning.
I hope this helps.