When buying used, the best thing to do is to "think old in years - young in miles". A 2010 with 100k miles on it is an OLD car. A 2000 with 50k miles on it is a YOUNG car.
And a car like mine, a 1993 Grand Am with 86,000 km (approx 53,000 miles) when I bought it? Young or old? Hard to think of a 20 year old car as "young," but it's certainly young for its age!
I don't know enough about cars to advise on make and model, but from my experience buying an older car, a few thoughts ...
You don't have to personally know cars well to know whether a car is in good enough shape to buy - I took a car I was thinking of buying to my mechanic friend for a once over. It cost me, IIRC, about $100 at the dealership where he works. He ended up advising me not to buy it because it would have needed a lot of repairs, but found me another car that the dealership had taken as a trade in but was planning to send for scrap purely on the basis of age. Yes, I had to pay again for safety certification and emissions testing on that car too, but at least then I found out that the tires and brakes were new, etc. There was no visible rust at the time (I've found one spot since then) and I believe it had probably been parked in an underground parking garage - the former owner was an 80 yr old lady who probably only drove it to church and the grocery store! You wouldn't want to pay $100 multiple times, but when you've narrowed it down to a strong possibility, you can at least make sure you're not making a horrible mistake.
Also, let people know that you're in the market for a older car with decent mileage. I talked to one friend the week after they sold a great older Toyota with reasonable mileage to another friend for a great price. I still cry a little over that one, as they would have been meticulous about maintenance and I'm sure the car was in fantastic shape. Then I got a lead on another car from another friend (it was her sister-in-law's car). That's the car I took to my mechanic friend - at that point, he knew I was seriously in the market and went looking for me.
Now, cost/repairs. With a much older car comes more repairs, but at a much cheaper purchase price for the car, you may be able to afford them. I paid only $400 for the car itself. Emissions testing and safety certification, a few minor repairs, initial licensing fees, and a gift certificate to a local steakhouse for my friend (who had come in on his day off to make the needed repairs) - I think I drove off the lot for $1100 or so. Compared to the $10000 you're planning to spend, you could bank quite a few thousand to cover any repairs that would come up :-)
And then there is insurance. With such an old car, that cost me so little to buy, it was an easy decision to go with the bare minimum of insurance allowed by law. That saves me money on insurance every month, and if something ever happened to the car, I'd look for another similar car so I could keep my insurance costs lower.
Ok, so I'm not necessarily suggesting that you get a car as old as mine - the gas mileage isn't great, I'm transporting just myself and my dogs, not children, etc. So YMMV. But I am suggesting that if you widen your range of make/model, age, mileage and price, you might find a great car that saves you money in one area or another, and still gets you what you want.