Here's my thoughts having bought several cheap cars for our kids, consultant for other friends looking to buy cheap first cars, and being a hobbyist mechanic.
Buy from private sellers, stay away from dealers of any kind.
Try to find a one owner, and ask about maintenance records.
Do not try to buy the popular cars, Civics, Camry's, etc.
Buy the ugly cars that no one likes (humbles the teenager too)
For a teenager get something at least midsize, they are generally safer.
And learned from dealing with people selling sub $2k cars from time to time, the very first question you should ask is, "Do you have the title and is it in your name?" If the answer isn't an immediate yes, hang up.
Second question is "Is the car registered, insured and inspected?" Be wary if not.
Not sure where you live, but look for cars being sold by individuals in well off neighborhoods. Most of the time, they just want to get rid of it and deals may be better.
Stay away from high mileage cars no matter what.
Use cars.com to check out asking prices for any model you might be looking at. Do a wide, like 500 mile search, then you can sort on year, mileage, etc. You can also filter private seller also.
Find the cars that don't hold their value well. We bought two Volvo V70 wagons for our son for around $3000-$4000 (this was maybe 8-10 years ago). Around 100k miles. They are very reliable if well cared for, and very very safe for young drivers. (and virtually unstoppable in the snow!)
Ford Taurus (don't laugh). They made and sold millions of them. Most are dead now, but you can sometimes find one being sold by someone elderly, or their family, with very low miles. Parts for them are plentiful and cheap (I bought an '02 Taurus wagon 15 months ago, have put 15k miles on it, paid $1300 for it.)
And just for more laughs, Buicks. Nobody wants them, I couldn't GIVE my in-laws LeSabre away. I think they have some newer non land yacht sedans now that don't sell well at all. (check cars.com)
Again, highest priority for me for a car in that range would be individual seller, one owner, has maintenance records, currently registered and insured.
I could also make a list of mechanical things to look for but not sure you were looking for that. Good luck. Buying cheap cars is a chore....