there are quite a few variations in web servers (Microsoft IIS, Apache, emerging "light" web servers) and hosting types (shared hosting, VPS, dedicated server, cloud computing)
upgrades need a strategy
backing up your site and how to revert if necessary
is your theme custom developed?
do you have custom modules?
how to upgrade modules and themes
are the contributed modules available for the new version of Drupal?
staging server for testing and testing the upgrade again before running it again
putting your site offline, coordinating with everybody including site users, readers and maintainers
running the upgrade
upgrading can be done in several ways
SFTP and phpMyAdmin for shared hosts
using Wget/Curl + tar commands and MySQL at the command line
CVS checkouts of core and modules
using the upgrade as a chance to setup Drupal in a sustainable way
multi-site install instead of multiple installs?
moving modules into their proper places (sites/all/modules, sites/example.com/modules as appropriate)
things to check once the upgrade is complete
all content and Views are properly upgraded
internal and/or external documentation is updated?
permissions are assigned appropriately