You don't need anything other than your qualifications, in the Valencia region I believe that to own a salon you need your certificates stamped by the local government (I'm mobile and haven't had to get any certificates stamped), but I'm not sure that the same applies elsewhere in Spain/Spanish Islands.
You need to pay 'autonomo' every month which goes towards your pension and healthcare, it's just over 200€, as well as the normal tax etc.
Vanneb lives and works in Tenerife, she should be able to help you some more and I'm sure she won't mind you sending her a private message.
hth's a bit