pick a target price or mileage that works for you and then find something accordingly. Personally I would be leery of any car under $5,000, and mileage is pretty wide open but I would be comfortable buying anything up to 100k miles if it was known as a reliable model and has had regular maintenance.
So another poster said he gets 70k+ mile cars for around $6k; I got a used economical car at 25k a few years ago for around $11k... I'd say it's pretty much a sliding scale between those numbers with some variation for how much the original car cost. Personally I wanted low mileage because I knew I would be putting on a ton myself, so that was my primary target.
One thing to be mindful of on higher mileage cars is when the timing belt is supposed to be changed. It's probably the most typical costly maintenance item that has to get done on schedule. Tires would be the other one (other parts that can go bad eventually, like the transmission, aren't as predictable/regular). People will often sell a car about a year before they'd have to change the timing belt or tires.
You can look up at what mileage any car is supposed to have its belt changed via Google, and then keep that in mind while shopping. Also any seller should be able to tell you how many miles are left on the tires (there are ways to tell visually if you know what you're looking at, but usually the tire maker provides a good estimate of the tire's life when sold, so if the owner has a receipt from the tire purchase you can also have a good estimate from that).
Also, buy Japanese for reliability. Or maybe some of the recent Fords.
I have a very short commute to work (1 mi.), so I generally have been buying economy cars that are between 6-8 years old with approx. 70-100k miles on them, running them for 3-4 years and then selling them. I generally pay $6,000 or less for them. I look for reliable, fuel-efficient brands and only buy from private sellers (no dealers) to save a few extra $$. It's also good to run a Carfax check on any potential used car.
Cars last longer and are better made than any time in history, so I always think that used cars can be a great deal.
I always buy used cars as well, and I only buy from private original owners. I try to find cars owned by people who took care of them. I ask to see the repair and maintenance records, ask how often they changed the oil etc. Sometimes it takes a couple months to find the right owner selling the right vehicle, but if you are prepared to do the legwork and keep the cash handy, it can be worth it.
As for makes and models, we've always had good luck with Toyotas and Subarus. We've never had a Honda but I'd look at them too. If you don't know what to look and listen for when buying a used car, ask the seller if you can let your mechanic inspect it for you.
We bought an old 4Runner for my daughter seven years ago when she was still in college for $5k when it had 150k miles. That car is still running strong and approaching 240k now with nothing other than routine maintenance. She's driven it from Pikes Peak to Maine to Quebec and several other places in between. The seller was a traveling salesman who put on a lot of highway miles (the easiest kind), drove the car through the car wash at least weekly to keep it presentable, and changed the oil monthly using only Mobil 1 synthetic.
I've done a few car searches recently, talked to my mechanic at length, and he says about the same thing. If you need something you can take out of town without worry, you might increase your minimum a few thousand and try to get something with a little lower mileage.
If you don't mind spending a little more, you can try CarMax. Some people just aren't cut out for negotiating with "strangers". People I know have had good experience with CarMax.