“Arrest the usual suspects”

Look underhood and underneath for marks indicating socket wrench removal of bolts, new gaskets, rtv, critter chew marks, etc
Run a soft rubber magnet along bodywork
Loosen a valve cover hose and watch/smell for oily blowby gas

Ask to take an oil sample, watch face during response
Ask to hook up an OBD-II scanner, watch face during response
Ask: what would a Carfax list, watch face during response

On test drive pay attention for vibrations/noise over range of rpm and speed
Find steepest hill where you can apply full throttle but stay below speed limit
Lift hands briefly from steering wheel to check for straight tracking
Examine all tires for different tread wear
Find near empty parking lot to do full range of left right turning
Apply brakes both straight and in curves

Do not ignore gut feelings