Drupal - How do I write automated tests?

http://drupal.org/project/simpletest - it provides essentially the same testing API as the core D7 module and you still have to write the tests yourself. There is also http://drupal.org/project/simpletest_automator that apparently will help you 'record' your actions on a site into a SimpleTest for you.

Additional helpful links:

• http://drupal.org/project/selenium
• http://drupal.org/simpletest

For site testing you may not find Simpletest the best solution. It's designed to test code, rather than the behavior of a site. You may be interested in more big-picture testing tools like Selenium, as Dave Reid mentioned.

A number of contributed modules for Drupal 6 includes simpletests. If you need inspiration take a look at the Date module for instance.