Most used cars have problems. You can get lucky, but don't expect everything to still be in good order. Even if you buy off a lot, from a dealer, you can expect problems. Use cars are great if you have the tools, the time, and a little know-how. Otherwise they may not be your best option.
Anyway, check the fluids for colors and odors. Transmission fluid is usually red, and doesn't get discolored over time, unless there are problems. Engine oil varies, fresh it's sort of amber, then goes to brown, then black... If it's black and thick, anticipate engine problems in the near future. Might also notice a rusty color, the odor of some additives, or even water/foam/froth, which might be a good reason to pass on this car. Radiator coolant usually green antifreeze, but can be different. Might be just plain water. A rusty color, or stuff floating around might not be a good thing. While under the hood, look for drips and rust stains. Look around and see if there are clean new parts, compared to the rest of the engine. Look for tool marks on nuts and bolts. Give you some idea if it's been taken care of, new parts are better then junkyard specials. Leave the hood up, and start the engine. Listen for unusual noises (turn the radio off!). Turn the wheels to on side, far as they go. Get out of the car, and look at the backside of the front tires, look at the brakes/rotor. Should be consistent wear on the tires, if not, might need some front end work, which I consider 'shop' work, usually need special tools. Rotor and brakes, look for wear and grooves in the rotor, how do the pads look?
Go back up to the engine compartment, look and listen some more, take your time, give it a chance to warm up some. Okay, now get in and back the car up about 5 or 6 feet, park, get out and look where it was parked. See any shiny spots, from leaking fluids?
Still interested? Take it for a test drive, pay attention to noises, vibrations, loss of power. Get onto a road where you can get up some speed, you want to make sure the transmission is running through the gears. After you've driven it, and all still is good, pop the hood again for another quick look. Should be a little hot under there, but not scorchingly high.
After all that, and nothing makes you wonder if this is a bad deal, it'll probably do for a while.
So far, the newest car I've ever owned, was 12 years old. I do almost all of my own repairs and maintenance. My youngest brother is into mechanical stuff, like I'm into electronics, so I get plenty of help on the more difficult stuff, but rarely need to ask.