Have a read through the HMRC site for self employment.
HM Revenue & Customs: Self Employed
Basically, you need to keep a record of everything you earn from being self employed, so use invoices and start a book or spreadsheet where you list all in your earnings.
You also need to keep a record of all business expenses. This was explained to me as anything you need to run your business. As a designer, it can be software, books, pens, business cards, web hosting, sticky notes, literally anything that you use whilst working. Keep the receipts for anything you buy, and all invoices for work you've done for someone else.
Eventually you will need to fill in a self assessment tax return where you declare your incomings and your out goings. If your earnings are below a certain threshold, you don't have to pay certain things.
You won't need an accountant until you're earning a great deal more, but many offer free consultations so you could take advantage of this and get some free advice. I also heard that HMRC offer a free drop-in service where you can pop down and get free, confidential advice about anything to do with your business so look into that too. You could try a citizens advice bureau too, if you can find one that's still open.