When it comes to buying a used car it can be a scary or intimidating process as you do not want to take on someone else's problem. I have personally bought and sold many cars in my years not typically for profit but because I love cars and for me there is no such thing as a perfect car. So I like to buy cars that I like at a good deal and then enjoy them and sell them before they lose value.
What you need
When you arrive to inspect the vehicle I would bring the following items:
Flashlight (so you can see any hidden or hard to see areas)
Clean rag (to check fluids)
Mechanic gloves (to keep your hands clean)
Obd scanner (to see if any codes have been cleared)
Knife (to protect yourself)
Pen ( to sign your paperwork)
Cash (payment for the car)
Become a detective
When inspecting the car it is like a mystery and you are the detective trying to find the current status of the vehicle and the vehicles past. First I would evaluate the person selling the car. I know stereotyping is bad however understanding the person you are purchasing the vehicle from might be your biggest clue if this is going to be a good purchase or a bad purchase. For example I would much rather purchase the vehicle from a older well groomed individual over someone younger who may not understand the importance of vehicle maintenance and care (I know I didn't at a younger age).
Check the body
So first things first I request the keys for the car and start the car and look to see if it has already been warmed up, sometimes this can hide cold start issues but there isn't much you can do about it. Then I turn on the a/c and let the car idle as I am doing the inspection. This action shows the cars ability to idle properly and would make any overheating issues known. I would then walk around the car and inspect every panel for any damage, nothing is worse than getting your car home and realizing its not as nice as when you bought it. After checking the obvious panels I check the underside of the car for any damage and the engine bay and door jambs. When checking these points its worth noting that you can tell if the car has had body work done by checking the jambs as they might have a fine line indicating a tape line where the car was taped off to paint a certain part of the vehicle, if they don't tape the jambs you might see overspray (build up of paint that is not smooth). When it comes to checking the front of the vehicle look at the bolts that hold the fenders on, they are painted on the car from the factory and if you remove them the paint will break off or be slightly damaged indicating the panel has been removed. Lastly on newer vehicles all the major panels have a sticker that will have a number that should match the rest of the panels, if the sticker is missing or a different number that panel was most likely replaced. This would also be a good time to check if the tires/wheels are in good shape and if the brakes look good as well.
Inspect the interior
Now that you know your body is in good shape I would check your interior, I would lock and unlock all the doors making sure each one of them is working. I would also do the same with each of the windows and power mirrors. Then I would check all the controls and switches on the dash, including climate controls, radio, traction control, towing, etc. You would also want to check your lights to make sure they all work and all the seat belts as well. At this time you can use your Obd scanner to see if you have any pending issue codes or if the codes have recently been deleted. Its important to take your time as any over looked issues you will be responsible for once the vehicle is yours.
Look under the hood
Once you have confirmed your body and interior are acceptable its time to check your engine and transmission. With the car running I would check the transmission fluid level and quality if possible (some new cars do not have an option as they are sealed).This should be at the high point of the dipstick as the car is warm, it should look mostly red, if the transmission fluid is brown or smells burnt or has debris these are all signs of neglect or a problem. Then I would shut the car off and check the oil, make sure it is not low and you would typically like it to look light to dark brown, if the oil is black it probably needs to be changed. It is very important that if you see any metallic debris in the oil or if the oil is creamy these are signs of a serious issue. You can also check the underside of the oil cap to see if you have any build up as this could indicate neglect or a possible issue. Next i would check the coolant. DO NOT OPEN THE RADIATOR CAP, this will blow off and burn you very badly. I would just check the overfill reservoir to make sure the coolant is at the proper level and if it is not a clear color such as green or orange and it has a milky look to it this could point to a problem such as a head gasket failure. I would also check the brake fluid and power steering fluid, usually these are ok unless its a older vehicle. Lastly I would check the windshield wiper fluid to see if it is topped off as this is a sign of good maintenance.
Take a drive
If all other aspects of the inspection are not deal breakers you are ready for the test drive. When driving you want to check the acceleration to confirm the engine is running smooth without any hiccups or noises also check the shifting to confirm the transmission isn't jerky or slipping. The transmission should have firm smooth shifts almost seamless aside from the engine changing rpm. Next you want to check steering, on turns make sure there isn't any squeaking or loose feeling from the suspension and you want to confirm you don't have any groaning from the power steering system. Other potential issues could be bad cv axles by popping or clicking or bad wheel bearings by a low hum or roar from a specific corner of the vehicle. As you are driving straight you want to confirm your car is properly aligned by feeling if the steering wheel is pulling to the left or right. Your drive should feel smooth and should be free of any thuds, squeaks, grinding or popping. Lastly you would want to check your braking to make sure it is smooth and doesn't vibrate the steering wheel or have loud squealing or grinding.
Talk with the seller
During the test drive you will most likely be with the seller so it is a good time to find out why they are selling the vehicle. This is important as a reason for selling due to chronic vehicle issues is much worse than moving out of state for example. I don't typically hit them with a bunch of probing questions but more of a good conversation about who you are and who they are with a few interested questions about the vehicle. Now that your done with the test drive and everything meets your requirements you are now ready for the negotiation, remember the more information you have the better, Good Luck.