In addition to a mechanical inspection, you should do the following:
Take a look: Make sure the body parts line up, the paint matches, the doors open and close easily, and the tires show even wear.
Lift the hood: Check under the hood for leaky hoses, worn belts, and dirty oil. Automatic Transmission Fluid should be clear and reddish, and not have a burned smell. Radiator water should have a light yellow or green color.
Take a seat: Turn the ignition key to accessory and make sure all of the warning lights and gauges work. Start the car and check all lights and accessories and make sure no warning lights remain lit on the dashboard. Pay close attention to the airbag indicator lights. If these lights fail to illuminate as you start the car - or stay lit after the car is running - it is a warning that the car's airbags are not functioning correctly.
Perform a safety check: Try on the seat belt and take a test drive to ensure that you are comfortable while driving the vehicle. Make sure head restraints, roof structures, and windshield designs do not interfere with your ability to see clearly. Test the vehicle at dusk or early evening to determine your comfort with the visibility provided by the headlamps. If you already have a child safety seat, install it to check for compatibility.
Hit the road: Take the vehicle up to 35-40 mph. Make sure shifting is smooth and the steering is straight. When braking, a pull to the left or the right could indicate a brake problem. The steering wheel should not shimmy at high speeds and cornering should be smooth.
Check the sources: Buying through the classifieds? Check the name on the title and match it to the name on the seller's driver's license. Many individuals disguised as private sellers are actually unlicensed, unregulated curbstoners, who may pass problematic used cars on to unsuspecting buyers.
Take the automobile to a reputable mechanic: If you're serious about buying the car, the cost to have a professional examine it far outweighs the cost of possible hidden repairs.