I'm currently on the public beta of El Capitan, how do I upgrade to the final version of the new OS?
You don't need to upgrade. OS X 10.11 GM and final release are the same according to Apple and once 10.11.1 is released both the GM and final release will update to it.
The safe way: Boot into Recovery HD, erase the drive, and restore from your Yosemite backup (if you have one). Then download the installer and upgrade.
I wouldn't recommend just downloading the installer and updating, as bugs could be carried over.