1. Paper and a pen Make notes about every car marking down the VIN number, asking price, mileage, car features, what you like, and what you don't like. These notes will help you in the selection process as well as in the final deal negotiation.

2. Printed copy of used car checklist

3. Small flashlight Whether it's looking underneath the car to check for leaks and corrosion or when looking under oil cap to see the condition of internal engine parts, a small flashlight will be very handy.
4. Small magnet

5. Paper towel

6. CD-disk and tape to check the radio

The same about badly corroded cars; even after repair the rust will reappear later. Take your time; normally it takes 30 - 40 minutes to inspect a whole car body all around.
Look alongside of the car. The lines of the car body should be perfectly straight. Look at the top image, this Mercedes-Benz hasn't been in an accident, and you can see the body lines are perfectly straight. Look at the reflection: it is perfect, no ripples. Try to check the entire car this way. Ripples indicate possible body repair - check the second image, the rear fender of this Toyota has ripples because it was repaired after an accident.
Look at the paint from a distance and from different angles. Does the color of different panels match? Look at the picture - the driver's door of this Corolla is repainted. You can tell that by different colors of the door and the front fender.
Check the paint quality. Does one of the panels (door or fender) look shinier than the rest of the car? - Then it might be freshly painted.
Check all the gaps between the body panels. All gaps should be of the same width and even along the entire length. If any gap seems too narrow or too wide, go to another side of the vehicle and compare. Is it notable different? - Then it's possible the car has been involved in an accident like this white Mazda in the picture. Does the car have lots of rusty spots like the on the image? - I wouldn't even look any further. Even if you repair these spots, the corrosion will reappear later. Plus, it's probably even worse underneath; brake lines, fuel lines and many other components might be badly corroded making the car unsafe to drive. To discover repaired corrosion spots try to stick the magnet at the most common places where the corrosion starts - around wheel arches, lower door panels, etc. If the magnet won't stick - there is no metal under the paint, only the body filler. Open the driver's door and try to lift it up on the hinges. The hinges should not have excessive free play.
Look for the trailer hitch. If the car was using for towing a heavy trailer, the engine and transmission may be excessively worn.

Have a close look at the tires. Are they of well known reputable brand like Michelin, Goodyear, Bridgestone, etc. or "no-name" kind of product? Are they all the same or different? Look at the tread wear. The new tires for a passenger car have original tread depth about 10/32 inch (8 mm). When only 2/32 inch (1.5 mm) of the tread depth left, the tire must be replaced. Also the tire must be replaced if there is any mechanical damage like cuts.

It's very important to check operation of the heater and the air conditioner. To do so, start the engine and let it warm up for a few minutes. Turn the A/C on. Unless it's very cold outside, the air conditioner should start working immediately after you switch it ON. Within a few seconds you should be able to feel really cold air blowing from vents. If this is not happening, very possible the air conditioner doesn't work. Be aware, besides the fact that the A/C problem might be quite expensive to fix, this also may indicate that the car has been involved in a frontal collision.
Check the heater. Try if all heater fan speeds work. One of the possible problems with the heater may be a leaking heater core that is very expensive to repair. If you feel moist air is coming from the vents with antifreeze smell, and windows become foggy when the heater is turned ON, that may indicate leaking heater core.

If a car has been flooded in the past, even if it looks and runs good now, it will develop many expensive-to-fix problems later. This is because water gets inside many components where it shouldn't be and eventually, the corrosion will cause a component to fail later.
Look for mildew smell, water stains on the seats, roof liner and door covers. Inspect door covers and trunk covers for possible signs of water level. If it's possible, look for moisture under the carpet. Check lower edges of the door covers for water damage, look for corrosion and water debris in the spare tire well. look for badly corroded metal parts under seats. Check all door speakers - water could damage the speaker diaphragm. Coffee-with-milk color of the engine oil or transmission fluid may indicate presence of water. History repost can also reveal flood. Be suspicious if the history report indicates that the car came from another state.

It's very important to verify that the oil changes were done regularly. Why is this so important? - it's a fact that if the oil was not changed regularly, the engine, even the most reliable one, won't last long.
For example, I saw the 2002 Nissan Maxima in need of serious engine repair at 35,000 miles (56,000 km). This, despite the fact that the Maxima's 3.5L VQ engine is one of the most reliable in its class. The reason was simple - the owner drove 12,000 miles (19,000 km) without an oil change.
There are also few things under the hood that may indicate that the car was maintained poorly:
Extremely dirty engine (like on the upper image), burnt oil smell under the hood, badly corroded battery terminals, oil leaks, are all signs of poor maintenance, though if the engine looks clean and shiny, it doesn't yet mean it was well maintained; dealers almost always shampoo the engine before showing the car to the potential buyers.
With the engine off, open the oil filler cap and look at the internal side of the cap and inside the engine. If the oil cap and the parts you see in the oil filler neck covered with thick black deposits, it may be an indication that the engine wasn't maintained regularly.
Check the engine oil. Unless it was changed recently, the oil condition is an indication of the engine shape. To check the oil level, the car should be on a level spot with the engine OFF. Locate the engine oil dipstick (usually it has a bright color handle saying "Engine oil").
Pull the dipstick out, wipe it off and insert it back fully in place. Pull it out again and check the oil level. The dipstick usually has two marks on it or simply crossed area (about 1 inch). Upper mark indicates "FULL", lower mark stands for "LOW". Normally, the oil level should be between "LOW" and "FULL" marks on the dipstick. If the oil level is lower than "LOW" mark and the oil appears very dirty, like in the left picture, it's a really bad sign; the car may have excessive oil consumption or the oil hasn't been changed for a long time.
Look at the dipstick itself. If the part of the dipstick that goes inside the engine is dirty and covered with black deposits, this may also indicate that the engine was maintained poorly.
Look for a smoke from the exhaust - the blue smokes indicates that the engine is worn and consumes oil. Black smoke means excessive fuel consumption.

Check the automatic transmission fluid on the dipstick. The transmission fluid should not have a burnt smell. It should be clean and transparent. Dirty transmission fluid indicates transmission internal problem.
Another indication of transmission problem is delayed engagement. It's easier to note delayed engagement after a car was sitting for a while: With the transmission in "P" (Park) start the car. With your foot holding down the brake pedal, shift to the "R" (Reverse) position. Almost immediately the transmission should engage - it feels like the car wants to creep backward. This should happen very smoothly, without a strong jerk.
Shift to "N" (Neutral), and the transmission should disengage. Now, shift to the "D" (Drive) position. Again, the transmission should engage without a delay - you will feel the car wants to creep forward. This also should be without a strong jerk.
If there is a notable delay (more than 1 seconds) between the moment you shift and the moment the transmission engages, such a transmission is either too worn or has some problem, avoid this car.

Try to drive the vehicle at each gear. All gears should shift smoothly and easily without any noises, jerks or shudder. While driving at the second or third gear, try to accelerate suddenly. The clutch should not slip. If you feel slipping (the engine rpm increases but the vehicle speed remains the same), the clutch may need to be replaced. Try to drive with acceleration and deceleration - there should be no whining or humming noise under any condition. All the gears should shift easily and noiselessly. Try to shift into reverse; there should be no grinding noise.

The vehicle should start easily even if it's cold. It shouldn't shake, make excessive noise, or smoke. If you see a cloud of blue smoke (see the picture above) at start-up be ware, it's the first sign of possible engine problem - avoid this car. During a test-drive look for anything irregular such as vibrations, noises, etc. Pay particular attention to check the automatic transmission. Test drive a car for as long as you need to get the feeling, try to drive on bumpy road and if it's possible, take the car on a highway. Try to accelerate - there shouldn't be any hesitation
Does the engine make any irregular noises? Any smoke?
Is the steering responsive?
Does the car feel stable at a highway speed?
Any shudder, jerks, harsh shifting?
Does it make knocking or creaking noises when driving over bumps?
Is it too "bouncy"?
Does the car hold straight direction or pulls aside? Is the steering out of center?
Any humming noise?

Don't sign a contract and don't give a deposit until you are completely satisfied with the car. If you have any hesitation be prepared to leave, you always can find another car or come back later. Consider to test-drive another vehicle of the same model to have something to compare.
If you find any problems and the salesperson promises "to take care of it" make sure to discuss in details - what exactly will be done. For example, if the car needs new tires, what kind of tires will be installed - cheapest possible or of reputable brand?
As a last step, have the vehicle inspected by a mechanic of your choise.