brake sysyem on 1999 Oldsmobile Aurora

how do you properly flush the brake system

It is best to open the brake fluid resiviour cap and suck the fluid out of the resiviour using a vacuum pump (Mityvac make one pretty cheap). Top up resiviour with fresh fluid. Then open bleed screw at each caliper one at a time and connect vacuum hand pump to the bleed screw, draw fluid out of each bleed screw until fresh fluid comes out of each bleed screw, keep an eye on the fluid level and keep topped up. After all wheels have been "bleed" depress the brake pedal several times before driving the car.