I've bought several domain names through godaddy.com. It's cheap, it works. No problems worthy of mention and decent customer service. Just don't go for any of the add-ons until you're sure you need them. You can park the name at godaddy or use it on a site elsewhere.
I just bought my domain name on Monday. Cnet recommends www.aplus.net
package, more punch for the money plus 24/7 tech help (it's amazing how
fast these guys are to reply to my every whim). You get 2 free domain names
with the package (check out cnet's site).
If you do not go with aplus.net, I then suggest you go over to www.godaddy.com
and sign up with them. These are the best two sites, in my opinion, out there.
On choosing a domain name remember to keep it simple and easy to pronounce.