from one person to another is a fairly straightforward procedure. The actual process varies slightly, depending on which high level the name is in (.com .net.co.uk etc.) but broadly speaking it involves the current registrant contacting the registrar and requesting a registration transfer to you, the new registrant. Make sure that the current registered person does this well before the renewal date.
Note that I haven't used the word 'owner' when referring to domain name registration - nobody ever 'owns' a name in the true sense. What you do is register the name for your use during a specified period - 1 or 2 years usually - and during that time you have the right to exclusive use of the name, but that's all. Fail to renew, and you'll immediately lose your right - the name will be available for registration by someone else.
As for getting your site seen - well, that depends. Google and the other big search engines will index it automatically, although the time taken to do this varies considerably. If the site's subject matter is unusual (you breed pink Andalusian ferrets) it will be returned high up in the search results, but don't expect that to happen if you run a Jennifer Lopez appreciation site.