I'm not a well-and-pump pro, but I have maintained one for a long time.
The first thing to check is the pressure tank. Turn off the pump and drain the water pressure down. Then check the pressure in the tank with a tire gauge. The pressure should be two psi below your pump cut-on pressure.
If your pump is set for 40-60 psi cut-on/cut-off, the pressure in the pressure tank (without any water pressure) should be 38 psi. As the pump runs until cut-off, the pressure should rise in the tank until it reaches the cut-off pressure (60 psi).
If the tank pressure is too high, bleed it down to 38 psi. If it is too low, air it up with a bicycle pump or portable air tank or compressor.
If it won't hold the pressure just sitting there or if water blows out of the air valve, the bladder is shot and you need to replace the tank.
There should be a check valve on the suction side of the pump if it is a shallow well (25' or less), but that should only affect the surge when the pump kicks on (which may not happen if the check is leaking).
Here are two excellent pro well-and-pump sites for future reference: