as a dancer i'm starting private lessons to learn aerials and back handsprings. what should i expect in my first private lesson? What should i wear?

Where is the lesson taking place? At a dance studio or at a gymnastics gym?

at a gymnastics gym.

In that case I would wear a sports bra, a tank leotard or fitted tank top and fitted shorts. Bring a pair of socks with you. Different gyms have different policies about bare-footedness. Make sure your hair is very secure and out of the way. The first lesson they will probably be testing what you can do already. They should test your strength and flexibility. Gymnastics classes involve a lot of body conditioning. More so then any dance class I've ever taken. After doing a warm up and testing your strength and flexibility they will probably have you do a ton of basic skills like rolls and cartwheels, walkovers (do you have a backwalkover already?) and handstands and stuff like that, to see where your strengths and weaknesses are tumbling wise. Then they may spot you on back-handsrings, possible on a trampoline or an inclined mat to see how far along you are and to gauge how long it will take you to learn the skill. Then they will probably start teaching you some drills and exercises that will help you with the strength, flexibility and timing of these tricks. I don't know how long your class is, so that might not all happen the very first day but that would be my guess. If they do go straight into spotting you on these skills without working on strength, flexibility, drills, etc. then I would find a different gym.