I have been reading the quickstart guide about From API's: http://drupal.org/node/751826
I understand how it all works, but don't understand how to save the data entered into the forms into my own database.
Is there a similar page to the form api quickstart showing how to integrate it with a database?
In the link you gave us go to the section "Submitting Forms", you have in there an example to do some insertions in the database.
You can learn more about the SQL management in Drupal 6 on this page > http://api.drupal.org/api/drupal/includes--database.inc/group/database/6