How does it sound?
Listen out for any excess noise from the engine, windows, and inside the cabin. If there's a sunroof, open it and listen to the difference.
What happens when you accelerate or turn on the air-conditioning?
Does the car feel powerful enough when you accelerate? Since air-conditioning uses power, do you notice a change when you turn on the air-conditioning?
What are the brakes like?
The car should slow down in a straight and controlled manner. And, when it's safe to do so, put the car to the test with different braking actions (sudden, soft etc.).
Does the transmission feel smooth?
If testing an automatic, are there any jolts during the upshifts and downshifts? If the car is manual, does the gearshift and clutch action feel seamless?