Not nooblet at all!
To a first approximation, all you have to do is move the files and, if necessary, change the configuration of whatever web application like WordPress you're using on your old domain.
All the files for the website subdomain.mydomain.com are in the directory /home/cherryjello/subdomain.mydomain.com/ .
When you register or transfer newdomain.com, add it as a fully hosted domain in the dreamhost control panel. This will create the directory /home/cherryjello/newdomain.com/ .
Then all you need to do is copy the files and make any necessary configuration changes. You may also need to change any not-so-generically-coded url rewrite rules you have in your .htaccess file, but it's unlikely that you have any so you can probably just ignore this sentence!
Are you hosting a WordPress blog at subdomain.mydomain.com? forum? wiki? Tell us and we'll be able to give more specifics! After we find that out I may have a few more mySQL questions as well.
Free unique IP and $67 off with promo code [color=#CC0000]LENSFREEIP67[/color] or use [color=#CC0000]LENS97[/color] for $97 off. Click here for more options