Buying a used car is mostly about removing surprises. Before you talk about price, spend twenty minutes on the vehicle itself and the paperwork behind it.
Start with the logbook. The chassis number on the document should match the number on the car, and the registered owner should be the person you are dealing with or a dealer with clear authority to sell.
Then look at the car cold. Start the engine from cold if you can — most faults are loudest in the first minute. Listen for knocking, watch the exhaust for smoke, and check that the temperature gauge settles where it should.
On the test drive, find a quiet stretch and let go of the steering wheel briefly at a steady speed. The car should track straight. Brake firmly once. There should be no pulling, shuddering or long pedal travel.
Finally, ask for a service history and an independent inspection. A seller who is confident in the car will not object to either.