I'd recommend you buy a 10-15 year old car with relatively high mileage AND that has had the most common mechanical problems already fixed. That way, you won't take a beating in depreciation or repairs for your 3,000 miles/year. Most people can't tell the difference between a 2000 and a 2012 year vehicle anyway. You should have a lot of choices between $5k and $10k. I've owned over 60 vehicles and all were under $9,000 except for a $10k Dodge Daytona a couple of decades ago. Most were under $7k.
My wife and I currently own four vehicles: 97 Subaru w/188k miles, 2000 Subaru w/128k miles, a 2001 Dodge 3500 diesel with 261k miles, and a 2009 Kawasaki Versys with 37k miles. All are dependable and my total cost was $22k. I would not recommend a Subaru Legacy like mine for you because the gas mileage is awful, but if you live in a snowy area, the AWD can come in handy when two-wheel drive can't get you out of the driveway.
One of my training buddies drives a BMW convertible and a Lexus SUV. You'd think he had $60k tied up in cars, but it's actually less than $20k.
The main thing in buying used is to get the car checked out by a good mechanic AND to make sure that any common problems with that make and model have already occurred and have been repaired. If you follow my suggestions, I'd suggest you stay away from Fords, Chevys, Dodges, and instead look at Lexus, Toyota, etc. The better cars have been more likely to be serviced regularly.
If you're not too picky, I'd suggest you find a used car dealer who will buy a specific model car for you at a dealer's auction and pay him a finder's fee of $500. You'll save one to two grand on a $7k car that you can use to make any unexpected repairs. I've done this maybe 20 times. I'm positive that I've bought more lemons buying from individuals than buying this way.
If you drop down to $2k, I'd suggest an early 90s Saturn (with the timing belt replaced) or an early 90s Lexus... the little ES250, for example. I had one of these and loved it. I sold it for $1,800 and wish I had it back.