Maybe this is just me, but I would not consider leasing a car for a young driver. The cost if the car is damaged may be unreasonable.
And why muse about what the cost would be for the first 24-30 mos if it is leased. The OP said the budget was 10K or less.
Try car rental places like Hertz. They are constantly rotating their inventory and so sell cars that are less than 2 years old. I have known people to buy cars there. And you know they will be better maintained than a private seller.
RE #21 I think that depends. The person we bought our D’s used car from was fastidious with maintenance. We are fortunate that my husband knows a lot about vehicles and can tell a really well maintained car from a poorly maintained one.
No disrespect, but rental cars are “beat up” (clean version of saying it) like a taxi cab in NYC and generally poorly maintained. And when looking for a used car from a private party, just ask the seller for maintenance records. If they have few or no records, you move on after the initial contact. Or buy certified used (usually lease returns) from a dealer and pay a small premium.
I think people abuse rentals.
“I worry about safety a lot.”
Me, too. The old Car Talk radio show always recommended used Volvos for that reason as does the IIHS.
Agree with rental companies. Lots of joy rides from the younger sect.
Some will cringe but this will go to 200 - 250,000 without thinking about it. Just have to make sure the trans is good.
It may depend, then, on which rental place you are looking at to buy from. The people I know that have bought rentals have purchased cars with no dings or scratches, with warranties in place, and with regular oil changes. Of course this all can be true of buying a used car anywhere. It is just that people sometimes forget or don’t know that rental places can be a place to get a good used car. They offer financing and some are priced such that lower middle class people can afford them.
Prius batteries do not have a high rate of failure. But old Honda hybrid batteries do.
Rental car agencies have people on-staff to perform minor body repairs and paint. They charge the customer or the customer’s insurance company, if there’s damage. But the damage is still there underneath the fresh paint.
And any damage/future repairs during the rental car’s in-service can take several years (e.g., worn transmissions, axles/axle boots, suspension parts, etc.) to show problem signs, well after any warranty period.
But there’s always a chance you get lucky. 
If buying from anyone check the maintenance history. I bought a used car years ago (midlife crisis at 40) and the senior owner only drove it 6,000 a year or something like that .I went to the service dealer and he showed me all the records and said I would be stupid not to buy it! Ended up being a great car.
Looked at used as well as new recently, and found a lot of nice used vehicles at the local dealership. Take a look there as well as at CarMax.
If buying used, then always run a CarFax report. It’s not foolproof, but it’s a beginning of the purchase process. And typically much if not all the maintenance history will be included. Dealers can also run the maintenance history for you
I’d also add that I’d personally avoid any vehicle from the East Coast, Rust Belt or Gulf areas. Rust develops quickly, especially on the exhaust system, bolts, nuts, and suspension.
I drive a former rental car myself – Hyundai Elantra, which I absolutely love and which has been absolutely trouble free since I bought it about 3 years ago. Cost roughly $14K. It was an Enterprise rental, though I actually bought it from the local Hyundai dealer - probably a nicer car than I would have bought for a college student. I think there are pros & cons to buying a car with a history as a rental — the pro is that tend to be very recent model cars offered at good prices.
@sushiritto I would add the northern Midwest, and the Rockies to your list, anywhere that gets snow on a regular basis and then uses salt to treat the roads.
Should be able to get a Honda Civic (2014 or 15) for $12-13K. Great reliability and longevity…
https://www.consumerreports.org/cro/news/2014/02/honda-civic-hybrid-battery-reliability-gets-worse/index.htm is an 2014 Consumer Reports news article about what they found in their reliability survey for Toyota Prius and Honda Civic hybrid battery failure rates. While they said that “the Prius remains among the most reliable of all cars”, some then-recent years of Civic hybrids showed 30% or more of responding owners reporting battery replacement.
(Yes, the title on that page does not match the article.)
Make sure to check car complaints dot com for the year car you’re interested in. A highly rated car (i.e. Toyota Camry) can have a bad year (2007).
Hyundai Elantra or Sonata. Very reliable.
$10k is a reasonable price limit for a reliable used car. However, you’ll likely find lower prices outside of Car Max, including going through private sellers. There are many search engines online. Cars.com can be a good general feeling of what’s out there. Of course, there are some additional increased risks through buying a used car through a private seller. An inspection from someone you trust is important.
Also consider cost of ownership, not just up front cost. Sum of Insurance, repairs, maintenance, and gas is likely to very different for different models, and in many cases exceed the <$10k initial expense after a few years.
I bought a used Toyota 4Runner 4wd for mine. Great winter driver, can haul college stuff back and forth, will easily go 300,000 miles with good care. I drive old Volvo’s but h and the kids drive old Toyota’s. Both big old Toyota’s and Volvo’s are safe cars also.