If you´re new to blogging and you would like to have your blog installed in your own domain, first I recommend you learn more about domain names . You can check my archives for older posts about this topic or do your own research.
You can find a small logo of my favorite registar (affiliate link) in Night Clicks right side bar under "Web services I use" label.
After selecting and registrating your domain name you will have to think about website hosting. If you live in my country, you will need a credit card for this process. There are many companies offering hosting services so it won´t be that easy to pick just one as the best. I suggest you compare prices and the features offered in each plan carefully. If you´re just starting out, a simple plan will suit your needs and some plans come with a free domain name included. I get this sort of information on a few different webhosting directories. Webhosting Directories provide information, ratings, reviews and articles about this topics. If you click in any of the links in this post, you will be directed to one such directory. You can also check my archives for more directories options as I´ve written about this topic on several occasions.