Hi - as Jane says, you can buy domain names through lots of different companies - some will give you free space, others will do it cheaper, but then you have to pay for space for the website too.
I am a website designer, and will be happy to give you any advice you want - just PM me.
I tend to buy domain names from
www.1and1.co.uk because they are cheapest Ive found, and you can then redirect the name to any other hosting company.
However their hosting is not too bad and starts from about £1.99 per month and for that you get up to 10 free .co.uk addresses, free software and also a free website designing package (up to 8 pages) - with a sort of wizard template thing, so you can build it yourself. Its great for beginners, but again, if you get stuck please dont be afraid to ask me for some help.