It's a 'pressure on demand' water system. The pump senses lower pressure when a faucet is opened and pumps water until faucet is closed and pressure can build again. It then shuts off until the next time. Sometimes the check valve that is between the pump and the water tank will leak, causing pump to detect low pressure as water leaks back through the check valve into the tank. Often the check valve is built into the pump or right where line from tank enters the pump.
There is usually a strainer or filter in the line between the tank and the pump. Have you ever removed it and cleaned it? Also, bad water flow is often the result of a clogged screen in the aerator in the faucet. Unscrew, take apart, and clean it out. Check that any outside shower or other valves are in the closed position.
Usually the only air in the system happens when first filling water heater after it's been drained for winterizing. In that case, you reset heater bypass from winter setting to in use setting and turn on hot water to fill tank, push air down line and out faucet. It will bubble and pulse until air is flushed out, then go to steady stream. If you have air continued after that, you might have an issue with a valve partly open to allow air into system.
Bob & Donna
'98 Gulf Stream Sun Voyager DP being pushed by a '00 Beetle TDI