Sorry, I can't ignore this:
We are firm believers in the used car market.
..
We've never bought a used car
My guess is if you have bad luck with a lemon, you won't be so firm.
Anyway, you haven't said how used you are looking, so strategies will vary. I think you'll get a fair (but never great) deal from CarMax. I tried selling a car to them once and they found some problems with it, so I know they really do look at cars they get, unlike many dealers who just use the trade in as leverage for a sale. But I don't think you'll find a car for under $10K at CarMax. And not to say a car from a dealer is bad. I got my best used car from one. And my worst two--one was back in college, always a risk at that price, but the other was a nice looking 5 year old '90 Acura.
I think the best deal to be had is to find a car coming off lease. If it's from a friend, all the better. People may trade in cars that are in good shape, but sometimes it's to get rid of a problem. Off lease cars generally come back on a schedule, and while they could also have a problem, people usually planned well in advance to get rid of it at that time, no matter what. The pay out price of a lease can be a good deal, so getting it directly through the lessee is best. But if you have a relationship with a salesman and ask them to be on the lookout for a model coming off lease, it's probably worth it. Especially if you can see all the service records.
In any case, have you Kelly Blue Book app (maybe there is a better source now?) ready to look up fair values. Private party sales are
usually better deals, but again you may be buying a problem car, and you won't get any kind of warranty.
We have an biweekly email in our community, where people can post things for sale. Every month or two there is a car, and almost always the asking price is higher than what you should pay for at a dealer. These are mostly retirees, so maybe they really are "little old lady" cars, but I can't believe the prices they are asking. You can't just assume those will be good deals.