Domain registrar GoDaddy drew a lot of bad publicity for supporting SOPA, resulting in a large loss of business and a reversal on its public position. But wherever GoDaddy stands on SOPA, it remains one of the worst places in the world to host a domain. The company's terrible behavior and rotten customer service, combined with its awful, stupid, sexist ads make it a great candidate for "that company I'm so glad I'm no longer hosting my domains with."
Wired has a great HOWTO for switching away from GoDaddy to a variety of its better competitors (I use Hover).
First of all, choose your new domain name registrar and management service. There are plenty of services out there, including Hover, Ghandi, DNSimple, Namecheap and many others. Some mention a transfer fee of as much as $10 (Hover) and as little as nothing (Namecheap). In most cases that fee isn't for the transfer itself, but for a one-year extension of the domain registration at the new service. (In Namecheap's case, you'll pay $4.99 for your first year of registration.) DNSimple even courts frustrated GoDaddy users with their "Goodbye GoDaddy" promotion, which offers transfer "at cost." That means you'll pay whatever GoDaddy was charging you, starting at $8.50 per year.
Most of these companies have their own articles that explain how transfer domains to their service. The directions are essentially the same, but each site presents them differently. Some have short directions without screen shots, others get into more detail and provide images for every step. For the noobs out there, the screen shots are very helpful.