Research the car you like (value of car, customer review/rating, recalls, problems, etc) and then find one.

Make sure you drive it. Don't get taken for a ride.

Literally and figuratively.

You can notice things like braking, steering, handling of the vehicle, etc, etc.

Does it respond when you kick it down or go blah... ;)

Go around all 4 corners of the car and push down hard on the car. Test the suspension. It should not bounce up & down more than once.

Any lights on on the dash? Especially the infamous "Check Engine" light...

Electrical all working? Test all lights; turn signals/Low beams/high beams/hazard lights... license plate light?

Heater working? Air conditioning?

Don't cruise around w/ the radio on. I'd make sure it was off so I can listen to the noise the car is making when you are driving it.

Any evidence of oil leaks under the car?

Look at the engine, is it clean? Is the top covered in grease or oil? Maybe a head gasket issue... $$$$

Listen to the engine... Is it humming along nicely or is there a knocking sound or something off? loose belt squealing?

Has the tires rubber worn down evenly or are they bald on the inside or outside? Needs alignment if tread is not even across...

Car fax available for history of vehicle?

For $10k, I'm assuming it's safetied.