Go to Namecheap, Godaddy or Dynadot and register your domain there. You can pay via Paypal or credit card. Domains can be registered for minimum 1 year to maximum 10 years.
To buy the new domain.follow the below steps: -Go to Registrar like Godaddy,Name.com -Search for the domain name you want to register -Make an account on Registrar and make payment using Paypal or Credit card -After you make payment,the domain will come to your Registrar account
If this domain is not available , you can look at whois information and contact the registrant directly and ask him to sell it for you.
Godaddy is my choice. Cheap, the control panel is simple and full featured and the customer support is solid.
Godaddy will normally give pretty good prices, but also check for promo/coupon codes for it too (google)
I suggest Godaddy too as I use it myself and find the it very easy to use. Buying a domain name with them and also chosing a good and cheap hosting space is very easy too. But if you plan to host a large number of web sites in your hosting space then you may find Hostgator more beneficial. If you are new then it is important to keep things simple by having everything (domain/hosting) in one place.
Godaddy is a great registrar imo, especially if you want to buy multiple domains at the same time. Since they offer special discount then.
go to websites like godaddy , sqaurebrothers or bigrock and search for domain that you need. If its available go ahead and register once you register you need to pay using paypal or credit card after you are done with registration search some good hosting provider and host your site