Make a database and a user for it in cpanel, which should also give you a folder. Get the 1.5.15 Joomla Full Package and extract it in that folder. Navigate to the folder in your browser, and it will walk you through the setup.
The only thing that can be tricky is that when it asks you for the database name and user, depending on the host, you may need to prefix it with your cpanel id, but not always. This would be cpanelid_databaseuserid and cpanelid_databasename. Host is localhost. (Edit - no prefix for passwords; wasn't clear.)
Delete the installation folder per instructions.
Go to Site>Configuration and play with settings. On the Systems tab, set the Session Lifetime high so you don't get logged out during development.
That's enough to play with for now. I'll pick some good extensions out and email you a .zip later this evening to save hunting around time.