Upgrading your Drupal site's core software is essential to keeping our site safe. Thus it is important to know how to do it right. There is a great is plenty of documentation on the Drupal site. Just go there, follow the step by step instructions and say your prayers. Actually it's not that difficult but you do want to play it safe, do the back-up.

Here are the basic steps:
Back-up the database, sites directory, modules and themes.
Put site in offline mode
Create a list of modules that you have enabled
Turn off added and custom Modules, set theme to a core theme like Garland
Upload upgrade to server.
Upload sites directory, modules and themes.
Run script at www.site.com/update.php
Check to see if things worked properly
Reenable added, custom Modules and themes
Put website back into Online mode

Here is a video that walks your through the process.