You plan to buy a domain and you are looking a good company to make bussiness with? When buying a domain, you essentially rent it. That is, you pay an amount of money which gives you ownership from one to ten years. The most common situation is when you pay per year and therefore renew every year.
When you buy a new domain, make sure the price is low (it should be less than US$10 for international domains). Established companies such as Network Solutions (often written as NetSol in forums) charge around US$35 for the same product.
When you buy a domain, make sure the price is not strangely too low. In this case, check the fine print to see what the price is for renewals. Normally it is the same price as buying one.
When you buy a domain, you have the right to change registrar if you want to. There are specific rules to follow, however, essentially you can transfer to any other registrar. You normally do not have to pay for the transfer; you need to obtain an authorisation code from the existing registrar and give that to the new registrar. Both steps can take place from the administration pages of each of your accounts with the registrars. When moving to another registrar, your current registrar can be a pain by not acknowledging the transfer. Therefore, the transfer can take up to 5-7 days before a timeout is exceeded. You would need to contact their customer support to get them moving (hello mydomain.com).
As an owner of a new domain, you can have a list of free services such as URL redirection (forwarding all requests to your domain to a specific URL), mail forwarding to the e-mail of your choice (firstname.lastname@example.org), privacy options so that your personal details are not shown in a WHOIS query and more.
There are different companies offering domain registration services. I would recommend those companies that do registrations professionaly; customers would use them to buy hundreds of domains. Typically, the price for a .com, .net and .org domain is around US$6-7.
Check the comments of this post for recommendation of registrars.