You likely have a tank like one of theses below
http://www.do-it-yourself-pumps.com/airetainer.gif

One of those tips at the top is where you pump air into the tank like
you would on a car tire. The tank has a pressure rating on it
someplace. Inside is a bladder that is filled with air. This
bladder pushes against the stored water and this gives you water
pressure when the pump is off.

Anyway, what may be happening is 2 things. First the rise and drop of
the water pressure is when your tank empties and then the pump is
sending the water directly to the house plumbing. If need, you can
put air into the tank with tire pump. Just don't exceed the rating
on the tank.
Look for a number with PSI after it.

As for the hot and cold combining. Shut of the water flow into the
hot water heater. Then turn on only the hot water in the lowest
point in the house. This is usually the bathroom. Allow the water
to drain. If you turn the hot water on in the bathroom with the
water intake on the water heater off and you still get water, you
need to check your shut off valves.

You may have sediment in the lines, hard water, or just age has warn
out the rubber seals and washers inside the water valve(s).