If your vehicle is showing delayed engagement when shifting into drive or reverse, here are the most likely causes and the parts you'll need to fix each one.
Automatic transmissions rely on hydraulic pressure to engage gears. Low fluid levels or heavily degraded fluid cannot build pressure quickly enough, resulting in a noticeable delay when shifting into Drive or Reverse.
A restricted filter starves the transmission pump of fluid. If the transmission was recently serviced, an improperly seated filter or missing O-ring can cause the pump to suck air (aeration), leading to delayed engagement.
Electronic solenoids direct hydraulic fluid to the correct clutch packs. If a solenoid is sticking due to debris or failing electrically, it will hesitate to route the fluid needed to engage the gear.
Over high mileage, internal rubber seals harden and shrink, and clutch friction material wears away. This causes internal hydraulic leaks, meaning the pump has to work longer to build enough pressure to engage the gear.
The computer responsible for commanding the transmission may be failing or suffering from software glitches, causing a delay in the electrical signal sent to the solenoids.