1997 SW1 with Electrical Problems

I have a 1997 Saturn SW1 with a 1.9L SOHC and 125,000 miles. Recently it has begun to have electrical problems. Sometimes when I apply the brakes all of the electical instruments inside the car (radio, lights, speedometer, tachometer, gas gauge, etc.) Go off and come back on again. This also happens when I have the headlights on andswitch the high beams on or off. Other electrical issues are beginning to show themselves too, like flickering lights when I apply the brakes or put on a turn signal. Sometimes when I apply the brakes the lights come on brighter. Could this be the alternator (I hear that they often go bad on higher-milage Saturns) or is it a bad gound or something? Also, when my car is started and I release the parking brake the car shuts off until I pop the hood and fiddle with the battery cables. Is this related. And insight you have would be greatly appreciated. Thanks