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Updating your site often sounds much scarier than the actual experience is. In addition to the included UPGRADE.txt file, the online handbook has a great deal of documentation available at http://drupal.org/upgrade and a helpful support forum at http://drupal.org/forum/21. The most important step to remember is creating and testing backups of your site.
It cannot be stressed enough how important backups are when you are doing upgrades. This holds true for upgrading both Drupal core and contributed modules. You need to make sure you back up both essential parts of a Drupal site: the filesystem and the database. Every system can have a different way to do backups, so that won’t be covered in detail here. You can ask your system administrator or refer to the backup section of the upgrade guide on Drupal.org at http://drupal.org/upgrade/backing-up-the-db. Make sure that you test the backups as well, so that you are sure that you can recreate your site if something goes awry.