I saw that Drupal 7 includes some kind of testing framework (I haven't tried it out yet since my existing sites are still in Drupal 6 and likely won't migrate any time soon). How can you write automated testing for a Drupal 6 site, is there a module for this? How much can actually be tested automatically rather than requiring manual testing?
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:
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.