If you have never registered a domain name before you probably do not know the process, but, simply, you find, order and pay for the domain name via one of hundreds of reputable suppliers.
Simple, yes, but, as your details are generally available for all to see publicly (unless you use a cloaking service or, in some cases, register it privately), you have a greater chance of getting emails offering web design, SEO services and more. You may also get an email like the one I got recently, here:
This looks reasonably official and referenced the domain name and, to the casual user, could be considered official - luckily, Google did identify this as dodgy, but your email system might not!
Bottom line, is that, once registered, you do not need to do anything with your website/domain name except get it up and running. Then you can register it to search engines etc, but, there is not a thing called 'domain name search engine registration', so, please keep a look out for this and similar emails and please let us know if you have spotted anything and we will let the community know.