One of the things I really love about Drupal 6 is its "Status Report" page, located at "admin/reports/status" (or "Admin|Reports|Status report" if you're drilling down through the menu system). This is my first stop whenever I think something wonky is going on with a site.
This one page will tell you if the site has pending security updates, an out-of-date database in need of some update.php love, file security issues, and a bunch of other helpful color-coded directives. Perhaps one of the most useful thing this page will remind me about is the status of the "update status" module. I often disable this module during the development phase of a project (just to hide the annoying messages) only to forget to turn it back on when the site goes live. The "Status Report" gives me the info and confidence I need to know that the site is up-and-running with nothing obvious left behind.
Usually I'll find myself on this page after downloading an update to a previously installed module - only to find out that the update also requires me to run update.php to bring the database up-to-speed with the updated module.