Vehicle suspensions are comprised of many components which must interact with each other in a number of ways to control vehicle movement. Each one of these components represents a potential failure point, and all are vital to the vehicle's proper handling and safe operation. However, you don't necessarily need to know how every part of the suspension works to troubleshoot it; specific failures have specific symptoms, so all you really need to know are the symptoms to determine what needs to be repaired.
Drive on a straight and perfectly flat (preferably freshly paved) stretch of road and accelerate up to 60 MPH. Note whether or not the car pulls to one side or the other or whether the wheel randomly shimmies back and forth. A pull to one side could indicate uneven tire pressure, uneven wear on tires, bad alignment (due to normal wear, broken or missing components or shims), worn steering tie rod ends or a stuck brake caliper. Wheel shimmy could indicate uneven tire pressure or wear, a bent or out of balance wheel, a damaged tire, worn steering components or poor alignment.
Accelerate on a bumpy road from 5, 10 then 20 miles per hour. A car that regularly bottoms out or bounces hard enough to make you feel light in the seat is "porpoising." This indicates worn out or blown shocks although bottoming out can specifically indicate worn out or damaged springs.
Find a car park where you can get your car up to about 45 MPH, turn it in a 200-foot circle and execute a few quick left-right-left transitions at speed.
Drive the car around in the 200-foot circle at 45 MPH. If the car seems to lean excessively, you more than likely have worn out springs or a worn out anti-roll (sway) bar. If the wheel shimmies or seems to tremble as you turn, then you should either check the power steering fluid level, power steering pump or power steering rack. If your car cannot maintain the circling in a smooth way, check for worn suspension bushings and broken or bent control arms or tie rod ends.
Perform several quick left-right-left transitions at 45 MPH. If you hear a clunking in the suspension, you need to check the ball joints and control arms where they connect to the chassis. If the body of the car seems to pitch excessively during the transition, then you need to check the shock absorbers on the side the car is leaning towards. If the car's handling feels sloppy and loose overall, then check the steering components and rubber bushings for excess wear.