How to register as self employed online

HMRC website, I think you will need your national insurance number but it's relatively easy

www.gov.uk/set-up-sole-trader/overview

Read that page then go on to the next page and click the Register as a new business link which will take you to the registration pages. Read through each step, selecting "I want to tell HMRC that I am in business and need to register for a new tax or tell HMRC that the company is now active". Keep on reading the pages and you will get to a Government Gateway login point.
You are on your own from there but it should be fairly easy to follow through each step.
I would use your own name as he business name. I would provide a brief description of your service - ad hoc childcare. I would use Cash Basis accounting (this is the easy method where you record payments in when you get paid and payments out when you make payments).

I would suggest you read a basic book about accounting if you do not know about doing small business accounts. Small Business Accounting: The jargon-free guide to accounts, budgets and forecasts (Teach Yourself) Paperback – by Andy Lymer is useful.