Updating my very first install of Drupal
If you can use Drush, do. It makes life very easy. If memory serves it's as simple as issuing a drush up drupal command, and following the prompts.

If you can't use Drush then copying the core Drupal files over the top of the old ones is the way to go, but make sure you don't replace the /sites folder. On most production servers the /.htaccess file will also have been modified so make sure you don't overwrite it if there have been changes made.

Either way, make sure you take a backup of the code and database before upgrading so you can easily rollback if something goes wrong.

You may find the Update your Drupal site from one minor 7.x version to another minor 7.x version page a particularly useful read before getting started.

Also, as Kalabro points out in the comments, never run an update on a production server without having tested it on your dev/staging environment first.