Does FS2002 have any bugs in the auto pilots in the stock aircraft?  If not, I am wondering if I am using it wrong.  I will do a flight and have everything work successfully, land and then take off in a different aircraft, only to have everything go haywire (e.g., climbing to altitude, plane suddenly begins to climb too steeply and stalls.  Will not respond to anything, but just keeps trying to climb too fast).

Some settings will not change as you move from plane to plane example is the nav/gps switch if you switch it to gps then go to another plane it will still be set. All of the auto pilot settings might not transfer from plane to plane. Such things as speed and vertical climb rate as these differ from plane type to plane type. Also some auto pilots have a command switch that needs to be on where others do not. The best thing to do is to make sure everything is set as you move from plane to plane.
But if you jump from a B747 to A B727 there will be a initial jump in the air craft. You just went from a plane weighing 300,000 lbs to one that goes around 150,00 lbs. Weights are a guess only.
In over all it is not a good idea to change plane types in mid air. Try staying with the same type like going from one 737-400 to a different livery 737-400 the only differance here is textures. But avoid going from a b747 to a b737 or b727. 8)

Most panels have a switch marked NAV/GPS.  It is usually set to the NAV position to correspond with NAV HOLD in the autopilot.  If you wanted to fly a GPS flight plan you would leave the autopilot set to NAV HOLD and move the NAV/GPS switch to GPS.  Don't forget to turn it off when you reach your destination.