First thing I look over is the frame, make sure everything lines up, no cracks or signs of damage. Next, I look over the wheels to see how true they roll. I also look for overly greased or oiled hubs and joins. If it looks overly greased or oiled, wipe it off and test ride the bike, come back and look over it again to see if you can spot the oil drips. I once bought a bike with bad rear hub
Take the bike out for a long ride and test the brakes and shifters often. Keep your ears open.
Best wishes.