After inspecting with your eyes, it is time to take the car out for a spin to test out its mechanical bits. Remember to go slow as you get used to the car.
The first thing to do is to check that the engine starts properly without any prolonged cranking. Anything more than 3 seconds and you should probably walk away. Difficulty to start a car might be attributed to a weak battery or faulty starter motor. However, it can also be an indication of more serious engine issues.
After the engine has warmed up sufficiently, it should be idling around 850rpm. Disengage the transmission and get someone to step on the accelerator hard a few times and check that the car is not spewing any blue smoke.
Before you start the car, step on the brake pedal. After the car has started up, you should feel the brake pedal depress if it has power or served brakes. Step on it hard now and if the pedal sinks much further, there could be something wrong with the master cylinder.
During normal driving, as you brake, take note of any vibrations. This could mean that the brake disc could be warped and might need replacement. Listen for any screeching noise as well when braking, this indicates that the brake pads might be worn.
The first thing to test for an automatic transmission car is engagement time between the shifting and the gears engaging. It is normal for most automatics to have a slight delay when changing between gears. However, this delay should not be longer than 1 second. If the delay is long, it indicates a transmission problem.
To test, first step on the brake pedal and shift the transmission to "D" or Drive. The shifting should be smooth and not accompanied by any major jerks. Some minor jerking is fine though.
Now as you take the car out for a drive, check that the car accelerates smoothly and the gear changes smoothly and accordingly. If there is any shudder or delay in the shifting, it indicates that the transmission could be worn out. Also, if a gear refuses to up-shift on a level road, it could also mean that the transmission is faulty.
Also, beware of slipping. This means that the gearbox's torque converter is dodgy and the engine's torque does not transmit properly to the drive shafts. You will know this when you step on the accelerator hard and the engine rpm goes up but the speed remains the same.