For magento upgrades you will definitely save yourself a lot of stress by using ssh. So after your system->tools->backup be sure to manually backup your local.xml (usually located at magento-install-directory/app/etc/local.xml ) . The local.xml file which holds your settings and the info you will need to reinstall magento.
I cosign on the maurisource.com link in johnpearson's answer & the official directions, both are useful. personally I ran into an issue running mage getting this error:
magento-install-directory/app/code/core/Mage/Core/Model/Session/Abstract/Varien.php on line 125
but this link below helped. probably cause I was on a bluehost shared host box.
had to add this line to mage:
if you need to reinstall at any point during your upgrade. make a copy of local.xml, then delete it. browsing to your site will take you through the steps. open the local.xml copy to help guide you through install, be sure to use the KEY from local.xml