I do 'em for a living.
Everyone i have ever tanned has been a little bit embarrassed the first time.
Wear whatever you fancy. I'm shy so I wear a swimming costume with the straps tucked down. Most people wear old knickers, an old bikini bottom and maybe a strapless bra. Also nude is fine. I dont inflict paper knickers on my clients lol. The ideal thing to wear is the brief bikini that you bought and then got home and thought "bloody hell I cant inflict that on the general public".
De-hair at least the day before.
Leading up to the tan exfoliate and moisturise quite enthusiastically. On the day DO NOT MOISTURISE and shower with non moisturising soap or shower gel. Ideally dont wear foundation etc.
If you wear perfume or deodorant the guide colour (the bit that you wash off) will go green. BUT you do wash this off adter a few hours so not the end of the world.
Boobs are tricky. The only way to get some tan underneath large boobs is to lift them up, but hoiking them around is, as you say, a bit mortifying. Wearing a bra is a solution.
Wear loose clothes and flip flops to the appointment.
Phew. Sorry. Am going on a bit. A good beautician will tellnyou all this and more. I email an information sheet to my clients beforehand and text them on the day as a reminder and the next day to check they are happy. Most get right on their nerves lol