Hi Mike. Yes, you can buy a domain name for peanuts nowdays but you will need a server to host it on. Most hosts offer several different hosting packages with a free domain name. Which host you go with and what package you need depends on what you intend to use your website for. Hosts offer different packages with different storage allowances and bandwidth allocations so these need to be taken into consideration depending on what you want your website for.
Once you have your domain and hosting, you then need to set up and upload your website. How you do this depends on your level of knowledge of HTML or use of a website creating package such as FrontPage or Dreamweaver for example. Some hosts offer a simple to use web creating package online.
If you need any further help or advice, I'm happy to help.