There's pages and pages of information about how you can set up your business, and also helps with what you need to be aware of, how the company should be registered, what to tell HMRC and lots of other interesting stuff. Good luck.
aunty as island says, self-employed and sole trader are the same thing for an individual. Catch is that the Revenue don't like people being self-employed if they only work for one person/company.
Of course if you set up self-employed there's a very good chance that you might only have one client for a while, so the important thing is, are you willing to take, and available for, other contracts?
If you tell the Revenue (if they ask - they never asked me or my DH) that you only work for this person and hope to be employed by them at the end, they may have a problem and effectively stop you getting the work (unless your friend agrees to take you onto their books).
You should be fine with what you're doing but you do need to register as self-employed with the Revenue. You will need to pay your own NI. If you earn more than late 60s (I think that's the limit:check) you need to register for VAT as well. Look on the Inland Revenue website.