If you don't take it to a mechanic.....check the CV joints, front tie rods, clutch operation for slipping (if a manual), parking brake, smoke from tailpipe, creaks in steering, steering wheel vibration at speed, creaks in suspension over bumps, tire condition, take it thru a carwash and see if anything leaks, also check all lights, wipers, horn, muffler, leaning wheels, bald spots on tires, cracks at lower frame, damage to jack points, paint color differences, wear and bad smell in carpet...........and run a CARFAX.
Get it good and hot on the test drive. go back to the lot leave it running and open the hood...nothing should be smoking. both radiator hoses should be warm and not bulging....belts should not squeal.
2002 BMW 330XI
1988 Porsche 944