UNDER HOOD. Feel Fan Belts and Radiator hoses to see if soft and pliable. Fan belts should be soft, not frayed or cut, should move the thickness of one finger. Hoses should not have cracks near clamps.
DIPSTICK OIL should read between low and high when engine is turned off. Oil should be CLEAR. Metal flakes or particles = BAD.
Transmission Fluid: Check while HOT, idling in PARK. should read between low and high when engine is turned on. Smell and learn. If it smells BURNT, it is! Metal flakes or particles = BAD.
Visually check fluid levels: Brake, Power Steering, Radiator Water, Windshield water, etc. These levels will give you a hint as to how well the car was maintained.
Be suspicious if the engine compartment is Sparkling clean or recently steam cleaned. This often means seller is trying to hide something. Especially if combined with fact that car not in his name!
Look under Engine and Trans to see if Oil or ATF fluid dripping from Pan or Nut. Use white napkin to see what fluid is. Visually track source of leak and estimate severity.
Engine and power steering leaks have little consequence yet should be addressed so you know. Trans leaks have MAJOR CONSEQUENCE. RED FLAG! Different rules for different makes and models.
I advise keeping a quart of OIL and a quart of ATF in your vehicle so that IF and WHEN you feel your car is missing 5 or 10 percent of it's usual cruising speed / power, pull over at the next safe flat spot, check Trans ATF with engine running, check oil with engine off. Visually inspect coolant level in reservoir. DO NOT REMOVE RADIATOR CAP IF ENGINE HOT.
This practice is especially important when you first buy a car until you learn the rate of loss of OIL and ATF for your new car. Change oil every 5k miles. Consult Owners manual for ATF.
Change, or top up your fluids before long trips or steep hills like Santa Cruz Hwy 17 or Tahoe Hwy 80 which can overheat your car. Turn heater on to cool engine (most cars). Travel light. Be nice to your baby.
DO NOT REV YOUR ENGINE after just starting on a cold morning. This is the CRITICAL time when engine parts are NOT LUBRICATED. Cold parts wear more. Relax, chill out... Don't break it. Breathe.