If you are a Sole Trader with a turnover under £1,000 annually, then you do not need to register, however it is mandatory if it is likely immediately before you start trading.
However, there are some anomalies to this rule, for example; if you have losses within the business and the turnover is less than £1,000 you may wish to register to be able to carry them over to the next financial year.
https://www.gov.uk/set-up-sole-trader
If you are a Partnership, Limited Company, not for profit organisation or a charity you must register immediately, as trading commences.
https://www.gov.uk/set-up-business-partnership
https://www.gov.uk/limited-company-formation
You must register for VAT if your VAT taxable turnover goes over £85,000 or you know that it will. Your VAT taxable turnover is the total of everything sold that is not VAT exempt.
You can also register voluntarily if your business turnover is below £85,000. You must pay HMRC any VAT you owe from the date they register you. Therefore, you need to add VAT to all sales from this date. This is beneficial if you are buying from suppliers who charge you VAT, because this can be offset against VAT owing to HMRC.
You might also need to register in some other cases, depending on the kinds of goods or services you sell and where you sell them.
https://www.gov.uk/vat-registration