The first thing you need to worry about is how you're going to host the website. There's some free hosts around on the internet, but most people who want professional quality hosting use a payed host, as there's no banner ads, and it supports a custom domain (which is what you want). If your broadband connection supports it, you can also setup your own home webserver.

Once the site is hosted and setup, you can buy any domain you want (register.com for example), fill in the primary DNS and secondary DNS according to your web host's instructions.

You can choose to either keep the original domain or drop it, as it's possible to have more than one domain point to the same website.