[ What should I watch out for?
I had to think about this one a while, and concluded that it's too general a question without knowing a specific model and year. Even within a single model generation there can be variations depending on equipment and mid-year changes.
So as a general guideline things you'd rather not see are leaks to a drip from anything. Some things, of course a less critical while others should be cared for quickly. Take a rear axle for instance. It typically has something less than a gallon of lubricant capacity, so if it leaks it can lose lube to the point of overheating and damaging bearings before you know it. OTOH it's not usually all that hard to change a rear axle seal but you don't know if the previous owner DID run it that low. Same goes for a transmission, just that it can be more labor intensive as you'd have to remove the trans. No leaks whatsoever accepted from the brake system BTW.
Look at the coolant. Should be bright green on an ethylene glycol (antifreeze) equipped car, and red on a later car that has that supposedly 100K mile coolant (I forget the name, one brand is Dexcool). If it looks like rust, then the system has had a leak and run on water-only for a long time. And I do mean a LONG time, which means it'll be corroded everywhere inside. Look on the bottom of the cap if it has rust build up but the coolant looks good you know they changed it to sell the car but probably didn't fix the original problem.
Pull the oil fill cap up. Look on the bottom of it and down in the fill port. If you see sludge you know the rest of the engine is just like it, and probably worse. The problem sludge causes is that it blocks drain-back passages and the oil pick-up screen. This in turn starves the engine for lubricant and can lead to a seized engine in short order. In your price range I doubt you'll find pristine examples but try to keep this sort of stuff to a minimum.
If you look at an auto trans car, look at the fluid. It should be red. If its brown and stinks with an acrid burnt odor, then the fluid is burnt and can be an indicator of internal trans problems. OTOH, if it's brand new this can also be a warning sign that it was changed just to sell the car. Freshly changed fluid can be confirmed by looking at the pan and it's gasket. If the pan is clean (and nothing else is) and the gasket new, you know the score.
Start the engine cold. If you hear a rattle as the engine revs on cold start that quickly quiets, that is usually a sign of worn rod bearings and that you should probably head on down the road.
On an atuo trans car, select the appropriate gear for the situation and the trans should apply just like you're used to. If it seems to delay, then it probably really did and you should just mosey on to the next car. On manual trans cars, the clutch should engage smoothly and not result in a judder when you release it. That's called clutch chatter and requires the replacement of the clutch to fix. Stock stuff will be something less than a couple hundred bucks but the labor to do it is a pain, requiring removal of the trans. It can be done on a rear drive car in the driveway (BTDT) but better to avoid things like this if you can.
Take the car on an extensive drive if the owner will let you (you being young he may object. If so, have a parent take it out for you.) Put gas if you have to. Drive on the freeway for at least ten minutes to see if it overheats, starts making abnormal noises etc. Drive it in city traffic with the A/C on to make sure it doesn't overheat at low speeds. With an auto trans make sure it doesn't start slipping as it gets hot. Stop in at a convenience store for a few minutes to see if it's hard to start when hot.
That's about it, other than perhaps taking it to a shop for an evaluation. I can tell you right now, though, that they're going to hate any car in your price range and come up with a list of a thousand things needing to be done ( I know, I used to be a tech and did that to CYA, or rather C
MA
). It'll be your job to decide what's critical enough to sink the deal.
Once you have a specific model and year in mind, then perhaps some more specific info can be found.