I was thinking worn out suspension parts, too. Most people don’t realize they’re worn out until it’s too late…a little extra bouncing around can’t hurt anything right? Wrong. It can hurt a great deal, and have a tremendous impact on handling, steering and braking.
Bounce all 4 corners of the car. Just press down, let it up, and do it a couple of times until the car is moving pretty good. Then stop. It should settle right down, with no extra “up and down” motions. If it keeps on bouncing, then the shocks are worn, at least.
Wether it’s suspension or brakes, you need to get it taken care of. As Cig said, it’s more important than just about anything else.