The pressure gauge will slowly build up, to the cut off level- (which is set around 70) .
I believe the pump ( which is submersible) then turns off. BUT my gauge will then very quickly turn up 100( max on my gauge).
If I turn on a faucet, then my pressure gauge will very quickly fall to 70, and then slowly fall down to the cut in pressure.

I tried putting a brand new pressure gauge on- and the same thing happens.

I noticed that after putting the new gauge on,- that with the water turned off to the house, and the pump turned on (so to fill up the reserve tank)- the same thing still happens.

What is the problem- is it my reserve tank or my check valve to the well or what??

Let's start with some basics.

You should hear a "click" when the switch cuts off the pump at seventy pounds. When you hear the click, pull the little grey cover off and make sure the points are open (not touching).

Is there a filter or softener on the system?

Assuming the pump kicks on at fifty, then the tank pressure should be set to 48# (with the system off and the pressure drained to zero by opening a low level faucet).

A little bit of a wild stretch, but when you feel the tank after the pump cuts off, does the tank feel cool?

If I turn on a faucet, then my pressure gauge will very quickly fall to 70, and then slowly fall down to the cut in pressure.

To me that sounds like two problems. A ruptured bladder, resulting in a water logged tank and a leaking check valve.

Turn off pump and drain tank. Check tank air pressure. If you have no air pressure, pressurize tank and prepare to replace tank. You can temporarily use tank until replacement. Turn on pump and watch gauge. If pump turns off at proper pressure and gauge falls with no water being use then prepare to replace check valve.