Drupal 6 - create second contact form with redirect after submitting
I think that you need to install the Webform module:

http://drupal.org/project/webform

There might be ways to do this with the Rules module, but I think for overall simplicity, Webform is the way to go.

I'm pretty sure that the version for Drupal 6 allows a custom message to be displayed to the user. In fact, if memory serves, it also has the option to redirect to a particular page.