How to update your Drupal 6 core
Updating your Drupal 6 core is not as difficult as you thing, please not that updating means 6.x to 6.newx. Here are the steps to follow
1. Make a back of your site directory
2.Take a backup of your database
2. Download the latest release of your current Drupal version.
3. Extract the dowloaded version
4. Normally I delete the site directory in the extracted version so that you don't update the files unknowingly
4. Set your site on maintenance mode
5. Delete all the files & folders inside your original Drupal instance except for /sites folder and any custom files you added elsewhere. Before deleting the files see if you have done any modifications in the robots.txt & .htaccess files as you have incorporate them in the new intstallation files
6. Copy all the folders and files except /sites (assumed you have already deleted the directory in the newly downloaded folder) in to your original live Drupal version directory
7. If the update release includes changes to settings.php replace old settings.php in .../sites/default/ with the new one, and edit site-specific entries (eg database name, user, and password)
8. If there are any changes done in the .htaccess or robots.txt add those changes to the new files.
9. Login to your site as administrator
10. Run update.php by navigating to http://...yourdrupalsitename/update.php
11. Follow the steps to update
12. Disable maintenance mode
Your sites will be running with the updated version of the core now.