how do I purchase a domain name
 

I've never done this before

who am I supposed to be paying for the space/url?

you can get a free domain name for a year when you buy webspace with a company like powweb

if you only have a small site you are best off buying the domain name from 123reg.co.uk and getting the space somewhere else

it's probably going to be a single page

so I buy the domain name and then I can do whatever I want from there?

once you buy the domain name you will need to own some sort of webspace to point the domain name to

no point getting a big hosting package like powweb's though if it's going to be a single page