You want your own domain and you don't know how to do that. Here is how I did for larve.net.
You can find a list of registrars from ICANN.
What's the difference between the different registrars? Well, first the price. Second the quality. I must say that I'm not convinced that there is a good registrar out there, but I found one which was ok: TLDNames. I carefully avoided Network Solutions because they are expensive and I am not sure that they are very reliable. I chose TLDNames because a friend of mine recommended them and because they are French. Well, I'm French, and in case of problems, next time I go to the French riveria, I can go and bug in person them if I am unhappy.
I can't say that I didn't have any problem with them since it took one week to get my nameservers listed correctly and that their registration CGI is kinda buggy, but I don't really regret my choice.
You need at least two machines with a permanent connection to the Internet. You can probably find that at your work if you do not work for a paranoid company and that your boss is cool.
You need a primary and a secondary nameserver. You also want a mail server, and probably a web server. Basically, you can put the primary DNS, the mail and the web server on the same machine. If you find someone who wants to host your secondary nameserver, you can do all that with only one machine.
Once you have done all that, with little configuration, you get something running.
You can send them at hugo@larve.net.