This sounds like there is an accumulator tank installed. The water pressure from the accumulator tank is lower than the water pressure from the pressure pump. The tap is opened, and the accumulator tank begins pushing water to the open faucet until pressure on the pump side drops and triggers the pressure switch on the pressure pump. The pump begins to run and pushes higher water pressure.
The other hint would be that once you close the tap, the pressure pump continues to runs for a few seconds as it repressurizes the accumulator tank.
The other thought is if you have a variable speed pressure pump. It is supposed to alleviate the need for an accumulator tank. I installed one this year and hate it. Water pressure drops in the shower
and drives me crazy.