2001 Pontiac Aztek Transmission "slips"

When in drive and starting out from a stop, I press on the accelerator, the engine revs, but the car doesn't move. Then, something "engages, " and with a little bit of a jolt, the car starts moving. This seems to get better after driving the car for a little while. The transmission fluid level is checked and is fine (when car is running and warm). This problem has been slowly getting worse, and is now not engaging (sometimes) when moving and downshifting is needed due to acceleration. Ever since I've noticed the problem, I've been careful not to press the gas pedal too quickly, so that when whatever doesn't engage finally does, it doesn't slam in and jolt the car too badly.