The easiest way to buy a domain name is through your intended website provider. For people who want a no muss no fuss solution this is the easiest way to get started.
Things to keep in mind:
If you decide to switch website providers it is more work to switch your name if you bought it through your provider.
Registering a domain name automatically makes your name and contact information public, most people find it worth the extra cost per year, generally $10 to keep their personal information private.
How to Buy a Cheap Domain Name, by WikiHow
How to Register Your Own Domain Name, by by Christopher Heng, SiteWizard.com
So, you have claimed your domain name. How do you set it up so that anyone who enters that domain name goes to your website? If you bought your domain name from your webservice provider they will do this for you. If not you can checkout one of the services below.
Redirecting a Domain Name, by Networking4All
How to Point Multiple Domain Names to One Website, by SiteWizard.com
Using URL Rewrite to route domains to subfolders, by CBeyond Cloud Services