Registering a domain name is easy. There are several providers to register domain name. Some providers charge early and some you can pay for several years. If you want web presence you can register domain with one registrant and host your domain in different place, or you can do doth in one place.
For example you can register your domain in www.1and1.com and host at godaddy.com (or you can do both at one location)
Some of the popular domain registrant
You have to research about their price and contract terms before purchase a domain.
Once you register domain name, information has to propagate throughout the system and it can take 4-24 hours to be fully visible around the Internet.