Hi Michelle from one newbie to another!
You need to be clear whether you are part of the salon or renting space for your own business. Who handles the money? Are you "nails at salon" or "salon's nails" Without putting ideas into the salon owner's head that they haven't thought about - are you building your own client base who will follow you if you move to another location, or if you leave will the clients will have their nails done by whoever rents the table next?
You need to talk to the salon owner and find out what they are offering in return for your money. It is normal to hold your own liability insurance and it isn't dear. You also need to be covered for other salon workers having an accident because of something that you've done -like tripping over something that you've put on the floor.
You should also check out about commission. If one of your clients takes a service from someone else, do you get commission and the other way round. It's polite to offer commission and it should help you build your client base.
You should also find out about tenure. Does the landlord need to know about you and what happens if the lease is up? How much notice will you get if you have to move out.
Make sure that you keep your own client records and that your clients know that you have them. (Data protection and all that). Learn to take good photos of your work, put a watermark on them and upload them to Facebook. Your clients will love having their nails on Facebook and will do your own advertising for you by sending links out!
Get a professional looking Facebook page, keep all the personal stuff off so that you look established, not like you have a hobby business run from your bedroom!