How do I show a different homepage to logged in and non-logged in users?

Rarst’s solution is indeed the simplest way to do it and the Members plugin also has heaps of other advantages.

You can also do it without a plugin, by using the is_user_logged_in() function

More info on http://codex.wordpress.org/Function_Reference/is_user_logged_in and http://codex.wordpress.org/Function_Reference/get_currentuserinfo#Parameters