This does not constitute professional tax advice. We strongly advise consulting a tax specialist to find out your specific obligations.
If you're located in the European Union, you should register in the country that you conduct business.
If you're not located in the European Union but plan to sell there, you can choose in which EU country to register. We recommend registering in the country where you anticipate you'll have the most orders. Or, you can register in the UK or another English-speaking country to reduce possible language barriers.
You can apply for a VAT ID at each country's State Revenue Service or tax institution that's is in charge of VAT registration. For example, UK customers can find info here. In the UK, It usually takes a maximum of 14 business days to approve VAT ID applications.