To some degree I disagree with 10-year-old or older cars the owner didn't take care of any more or doesn't want to take care of. If you know how to use the internet and how to use car sites you can find cars that are older than 10 years and were taken well care of. For example, about 3 years ago I bought a 2002 Lexus SC430 for a summertime toy, the car only had 38,000 miles on it, and it was garage kept, plus all service records, the car has run superbly since I bought it as a 20 year old car! That car cost more than $2,000, it was a $12,000 dollar car being sold at a Lexus dealership that had serviced the car since it was new, and they had it on sale from $15,000.
I gave my grandson our 2009 Acura TL with 150,000 miles on it, it now has 170,000 miles on it, we bought it with 42,000 miles on it, all those miles it has never had a single mechanical repair.
High mileage cars can be tricky to buy, I wouldn't recommend buy a 150,000 mile car, but we gave it to him, and I was confident in the car that it would be fine for him for at least another 100,000 miles! But if you use the internet correctly you can put in the max number of miles you will accept, like say 125,000, and stay with historically reliable cars like Lexus/Toyota, or Acura/Honda, and if the car has been maintained correctly it should be a great car. At the $2,000 price level you will have to accept some body imperfections, and maybe some interior rough spots, just make sure nothing is major, minor scuffs and scratches are fine.
There is another trick to buying cars, this works for any make and model, all you do is once you find a car in your price range and miles, simply go online and enter the year, make, model and submodel if any, or engine size, and the put after that "was this a reliable car" hit enter. Or you can also search by asking " what were the best years for such a such car", and it will usually come back with the best and worse years for that particular make and model of car. Doing this helps eliminate some of the risks of buying used. This works for any brand of car, in case you can't find a Lexus/Toyota or Acura/Honda in your price range and mileage you might have to expand your horizons a bit.
The other thing is DO NOT TRUST Car Fax! Why? because Car Fax only records damages to the car if there was an insurance claim filed, if the owner paid cash to have it repaired there will be no record of it. But Car Fax will at least show a history of maintenance and repairs. Always try to get repair and maintenance records so you can tell how well it was maintained. If you are not car savvy then take any car you're considering to a mechanic, but they will charge to check it out, which could tear into your budget, or find a friend who knows a lot about cars and take them along.