I am looking for a free website that keeps track of which US cities
have direct flights to which other US cities.  Preferably Presented in
a list or table format and including both regional and national
carriers.  I would find such a resource handy as I find direct flights
just a fraction as exhausting as flights with even a single stop.  It
would also let me plan multi-city travel without spending all my time
changing planes.

Try OAG Travel Planner

http://tistrial.oag.com/reg.asp

It's not completely free, but you can get a free 15-day trial.  If you
put in two cities it will tell you all the flights (and if any stops
are included) between those two cities (both within the US and
internationally) for all airlines, the plane, travel time, etc.  Hope
that helps.  If you like it you could always subscribe via the
internet, but I think you can also buy a paper version and/or download
one to a PDA.