Hyundai Repair: Elantra AC/Heat problem, coolant leaks, coolant level

Question
I have a 2001 Elantra GL with 100,000 miles on it. The AC cools, and the heater heats, but only when the car is moving at a significant speed, or when it's stopped but the RPM is kept at about 2500.  My feeble brain figures vacuum leak, thermostat, or temperature sensor.  Have you any other suggestions/ideas?  Are there easy ways for a shadetree mechanic to determine whats wrong?

Answer
Concentrate on the cooling system.  The A/C will be disabled if the engine gets too hot.  So it's likely that the A/C and heat are caused by the same problem.

You'll want to apply the standard tools to checking the cooling system.
1.  Check the coolant level.
2.  Check for coolant leaks if the coolant is low.
3.  Check for coolant flow when the vehicle reaches operating temperature.  (This indicates that the thermostat is opening).
4.  Check for radiator fan operation.  The fans should come on after the thermostat opens and continue cycling as necessary as long as the vehicle isn't moving.