Well pressure drops when pump turns on
Lacking any details (feel free to edit your question and add them) about your well pump type (suction, jet, submersible) or location, this mostly sounds like an improperly pre-pressurized accumulator (pressure tank.) If the pump kicks on at 30 PSI and the air side of the tank is not set at 27 or 28 PSI, the water in the tank may run out before the pump gets going, producing symptoms as you see.

If you turn the pump power off and drain water from the system, the pressure on the air side of the accumulator should be 2-3 PSI below the cut-in pressure. That 2-3 PSI allows for water to be supplied while the pump gets going. If your pump is submersible this is an easy check. If your pump requires priming to restart, it's a more involved check, as you'll have to prime the pump.