Domain name registration prices vary, but should be $35 per year or less (the best deals are much lower, but read the fine print; make sure you will legally own your domain name).
godaddy.com is one of the cheapest domain name registrars available. GoDaddy will strongly encourage you to purchase your web hosting service from them as well. And there's nothing wrong with their web hosting service. But keep in mind that you do not have to purchase domain name registration and web hosting from the same company.
Many web hosting companies can handle domain name registration for you in order to speed things along. They often offer the best deals on registration because they are counting on you to sign up as a hosting customer as well. For more information about web hosting, see how do I set up a website?
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