Check the transmission and clutch.
The clutch pedal should not feel spongy. Also check that the transmission shifts smoothly and operates quietly.

Check the acceleration.
If the car hesitates during acceleration or feels sluggish, there may be significant problems with the engine.

Tips for Purchasing a New Car: Final Inspection

Take the car you've got your heart set on to a trusty mechanic who can put your car on a lift to complete the following final steps:

Check the undercarriage of the car.
Make sure there are no signs of oil leaks or rust damage.

Inspect the brake pads and discs.
Check the amount of pad material left on each wheel; uneven brake pad wear is symptomatic of significant brake problems. Also check the discs to make sure they don't have any deep grooves or gouges.

All that's left to do now is shell out your cash (it's painful to do, I know) and hit the road in your sweet new ride!