There are really two issues you are dealing with here, so I’ll deal with them one at a time.
You cannot change a domain name you have already registered. It is yours, like it or not, for the amount of time you registered it for (unless, of course, you manage to sell it!). There are no refunds available for that, so it is a “done deal” that can’t be undone. You do not have to renew it at the end of its original registration term, but you can’t undo the initial registration itself or change the domain name for that registration.
Assuming that you register “myownwebsite.com”, all you have to do is “Add A New Domain or Subdomain” from the Control Panel -> Manage Domains screen, and specify the new domain name. DreamHost lets you host as many domain names as you want without additional cost - hosting and registration are two different things. Each registration costs money (you do get one free with your hosting), but the actual hosting of additional names costs nothing.
Once you have registered (which costs) and added hosting (which doesn’t) for the new domain, you can just copy the files between the directories of the domains using the shell, or by just uploading them again to the new domain via FTP.
You could also just rename the directory mydomain.com to myowndomain.com via the shell or via ftp.
If your site is already built, you will, of course, need to correct any links and image links that are not “relatively” coded to account for the new domain name.
It’s just a matter of getting the files into the directory indicated by the “Add a new Domain” screen in the panel, so if you do not ever want to use mydomain.com, you could just set the “base” directory for myowndomain.com to “home/user/mydomain.com” when you Add myowndomain.com, and be done with it!
There are other choices, involving redirects and such, but since you are just starting out it is best to keep them straight, IMHO.