When you are buying a car from a private person, it is sensible to put a couple of things on paper. A purchase contract, like you are used to from a dealer, is not available. To still be able to draw up a contract, as a seller or buyer of a pre-owned car, we have made one for you, that you can download here.
The engine is the most expensive part of a car and probably the hardest to judge. A quick peek in the engine bay is interesting, but what can actually be seen here? Under the bonnet you can find clues that point to the weak and worn out parts of the engine. A weak part may require maintenance, but can also mean very expensive repairs, in which case it is advisable not to buy the car.
The quality of a gearbox depends highly on the previous owner(s). This means that you can find out what the general condition of a car is by just checking the gearbox. A poorly functioning gearbox can very well mean that the car in general has been treated badly. An important part which you should check in combination with the gearbox is the clutch.
When you buy a car, it is nice to have a clean interior, without spots in the upholstery and broken parts. However, these defects are easily solved with a little soap and water and some small repairs. It will become more of a problem, if the upholstery is ripped.