How do I get Drupal to use my themes page.tpl.php template for the front page?
It is page--front.tpl.php - note the double dashes. 

The cascading matches my understanding. page.tpl.php should be called if there are no higher priority page templates. You need devel themer to work this out http://drupal.org/project/devel_themer. It will print out on the screen the template matches.

Details are in the drupal theming guide. in this section:

page--[front|internal/path].tpl.php 
base template: page.tpl.php