<p>If I hit a page which calls <code>session_start()</code>, how long would I have to wait before I get a new session ID when I refresh the page?</p>

Check out php.ini the value set for session.gc_maxlifetime is the ID lifetime in seconds.
I believe the default is 1440 seconds (24 mins)
http://www.php.net/manual/en/session.configuration.php
Edit:
As some comments point out, the above is not entirely accurate.  A wonderful explanation of why, and how to implement session lifetimes is available here:
How do I expire a PHP session after 30 minutes?