The following is a "sufficiently paranoid" approach to upgrading Drupal core and modules. As always, when one is going down this path the first thing to do is put the site in offline mode from admin/settings/site-maintenance.

Then make a back-up of your site, existing modules and themes, and database.


cp -r drupal-site drupalsite-bak
mysqldump drupal6-site > drupal6-site.sql


Download the core modules, based on the suggested updates from admin/reports/updates. You might even want to turn this into a script;


wget http://ftp.drupal.org/files/projects/drupal-6.20.tar.gz
wget http://ftp.drupal.org/files/projects/admin_menu-6.x-1.6.tar.gz
wget http://ftp.drupal.org/files/projects/ctools-6.x-1.8.tar.gz
wget http://ftp.drupal.org/files/projects/context-6.x-2.0.tar.gz
wget http://ftp.drupal.org/files/projects/diff-6.x-2.1.tar.gz
wget http://ftp.drupal.org/files/projects/event-6.x-2.x-dev.tar.gz
wget http://ftp.drupal.org/files/projects/gmap-6.x-1.1.tar.gz
wget http://ftp.drupal.org/files/projects/jquery_ui-6.x-1.4.tar.gz
wget http://ftp.drupal.org/files/projects/notifications-6.x-2.3.tar.gz
wget http://ftp.drupal.org/files/projects/quicktabs-6.x-2.0-rc5.tar.gz
wget http://ftp.drupal.org/files/projects/rsvp-6.x-2.0.tar.gz
wget http://ftp.drupal.org/files/projects/safehtml-6.x-7.x-dev.tar.gz
wget http://ftp.drupal.org/files/projects/token-6.x-1.15.tar.gz
wget http://ftp.drupal.org/files/projects/views-6.x-2.12.tar.gz
wget http://ftp.drupal.org/files/projects/wysiwyg-6.x-2.2.tar.gz


Extract the new core Drupal, Copy settings.php etc over from the backup


tar xvf drupal-6.20.tar.gz
cp -r drupal-6.20 drupal6-site
rm drupal-6.20.tar.gz
cd drupal6-site/sites/default/
cp ../../../drupal6-bak/sites/default/* .


Extract New Modules. You might want to turn this into a script too..


cp *.tar.gz drupal6-site/sites/all/modules
cd drupal6-site/sites/all/modules



tar xvf drupal-6.20.tar.gz
tar xvf admin_menu-6.x-1.6.tar.gz
tar xvf ctools-6.x-1.8.tar.gz
tar xvf context-6.x-2.0.tar.gz
tar xvf diff-6.x-2.1.tar.gz
tar xvf event-6.x-2.x-dev.tar.gz
tar xvf gmap-6.x-1.1.tar.gz
tar xvf jquery_ui-6.x-1.4.tar.gz
tar xvf notifications-6.x-2.3.tar.gz
tar xvf quicktabs-6.x-2.0-rc5.tar.gz
tar xvf rsvp-6.x-2.0.tar.gz
tar xvf safehtml-6.x-7.x-dev.tar.gz
tar xvf token-6.x-1.15.tar.gz
tar xvf views-6.x-2.12.tar.gz
tar xvf wysiwyg-6.x-2.2.tar.gz


Run update, and clean up


https://drupal6-site/update.php


Go into Drupal administration, mark the site online.

admin/settings/site-maintenance

Check input filters on: admin/settings/wysiwyg .. and make sure that BB code isn't enabled, but all others are.

Check input filters on admin/settings/filters ... and make sure that Plain text, Line break converter, HTML filter, and URL filter are not enabled.

Don't forget to copy your robots.txt from the backup if you have made any changes.