How to perform automated testing of an entire site after core changes?
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