The transaxle will follow a different shift strategy when the fluid is cold, so harsh shifting is normal in that case. Usually a fluid exchange will fix that right up. If that does not change the behavior, you could have a number of things happening here - some are very inexpensive to fix (sensors), to something in the transaxle itself.
Couple of questions:
- Have you tried to shift from part to L or 2? Does it act the same way as when you shift to D?
- Does the transaxle shift from P to D differently when the car is fully warmed up?
- Does shifting from P to R, then to D always "fix" the slow engagement?
If the fluid change does make a slight difference, it might take more than a couple of fluid exchanges before the transaxle reacts more directly. Also fluid choice is critical - this transaxle uses Toyota Type T-IV ATF, which didn't have too many aftermarket vendors. If fluid was used that was not compatible with Toyota Type T-IV ATF, that would cause serious degradation of the clutch bands inside the transaxle. Same applies the opposite way - a car requiring Toyota WS or Dexron II/III cannot use Type T-IV ATF - cannot be interchanged or mixed.
Wrong fluid type or infrequent fluid exchanges can also cause a significant amount of deposits and varnish to form inside the valvebody of the transaxle. The valvebody controls how quickly the transaxle shifts from gear to gear, and how long to how each gear. A clogged or faulty valvebody and associated shift solenoids seem the most likely issue in your case, assuming that fresh fluid doesn't fix the car.
Another common trouble area is the coolant temperature sensor, if the temperature reported by this sensor is incorrect, the ECM will believe the engine is much cooler than normal and run a different shift program - keeps transaxle from upshifting too soon, but generally doesn't affect take-up speed from position to position. Mentioned it as the sensor can be easily checked for operation. with a multimeter and a hot temperature bath.
The shifter assembly itself could be faulty. The shifter moves a cable to actuate a shifter arm on the outside of the transaxle, if the cable is too loose or too tight, could car the arm to select something that is not Drive. Again, this doesn't seem likely, but can be easily checked out.
But what could be happening, give the age and mileage, is that the motor mounts (torque dampeners) are worn out or damaged, causing the engine to rock back and forth much further than normal, which could cause binding in the drive line. Along those lines, slop in the drive axles can make the transmission seem "slow" to engage.