Like camoscato said, you need to register with HMRC as a sole trader. They'll send you stuff so that you can access things through the online portal.
Make sure you invoice clients properly, so that you have a paper trail, and keep receipts for any and all business expenses. I maintain a spreadsheet with all my incoming payments by invoice number (each client gets assigned a unique number, so that both they and I can keep track). Set aside the appropriate amount for both tax and NI contributions (I use an ISA for this, so I don't get double-taxed on what I'll use to pay my tax).
By the end of an April, I receive the various end-of-March payments, so I can close out that fiscal year. (I am very lucky in that all my clients pay 30 days net without fuss.) I gather up all my receipts (making any paper receipts electronic, for easy storage) for the fiscal year's business expenses, finalise the spreadsheet and send everything via Google Docs to my accountant, who is worth her weight in gold, because I'm now full-time self-employed, and don't want the hassle. Plus, she gives me the assurance that it's all done right.
You can easily do it yourself, though. After moving over, I was in full-time employment and self-employed on the side, so I wasn't making enough extra to make the hours I'd spend getting the correct allowances worth it. Like all government-related things, a lot of the language is what-does-that-MEAN ropey, but it's ultimately quite easy to do on your own.
They give you like seven different ways to pay, and must do so by 31 Jan if filing electronically. So, you'll be working as a sole trader in the 2016-17 tax year, will file sometime after 6 April 2017, and will have to pay by 31 Jan 2018. At the same time, based on your 2016-17 return, HMRC will calculate two 'payments on account' -- essentially, you will prepay on the final 2017-18 amount. This will result in smaller lump sum due for 2017-18.
I have a mate who will tell you the following is absolutely bonkers, as he prefers to do everything at the last moment, but I pay straightaway. So although my 2015-16 tax isn't due until January 2017, I've paid that. Also, three months into the new tax year, I've paid my first payment on account (also due January 2017). My plan is to have the second payment on account (due July 2017) paid by done by October. That will give me six months to continue to set money aside, with the idea I'll be able to give myself a big, fat 'bonus' in May 2017 when I find out my final figure. (In addition, it means I have an HMRC return to submit with my US taxes by 15 June of any year.) My mate, on the other hand, prefers to have his money in hand for as long as possible and doesn't mind working in one fiscal year to pay for the previous fiscal year.
Either way works, so you can do whatever across the gamut you're comfortable with, so long as you're fully paid up on time.
That's all I can think of right now. Jimbocz on this forum knows about this stuff, too -- he might have many pence to add. There might be things I'm wrong about, but my accountant didn't say I was doing anything wrong and thanked me for being organised, so all seems OK.