Sounds to me like you have an old galvanized steel storage tank. This tank does not have a bladder in it it uses air in the top part of the tank for expansion. You need to drain the tank completely. shut the pump off first. Put a hose on the drain valve and run it to a drain and then open up the faucets in the house to allow air into the water supply system(you should hear the water in the tank gurgling in the tank). This will probably take about an hour to drain the tank. When the tank is empty close all the faucets in the house and return the pump to operation.
This should take care of your problem and should be done once or twice a year because the air in the tank will escape out to the house through the water, leaving you with no air in your storage tank for expansion.