As AcidPerm says, you have to think differently being self employed.
You clients are your clients, so you keep their contact details, clients records etc
It's also up to you do do your own marketing and get business in. It may be good to educate your clients that you are self employed, not just one of the stylists. You don't want to spend time promoting your business, just to find a client responding to your marketing has come in and booked with another stylist.
Your landlord might promote your service, but they don't have to.
If you start being told what hours to work, when you can take holidays or get paid an hourly rate - alarm bells should start ringing!!!
Unfortunately there are a few posts on here about employed stylists being called 'self emplyed' by their bosses to avoid paying tax, ni , sick pay and work place pensions.