$3,000 or less can be a tough price range in which to find a good car. Have you taken into consideration the cost of having a pre-purchase inspection done on the car, or at least have a mechanically inclined friend to take along with you? Either will greatly increase your odds of not buying someone else’s headache.
As for cars, I will go against the grain due largely to your price range. Hondas and Toyotas generally get rave reviews and tend to be highly recommended, but for those reasons I would rather buy a $3,000 GM or Ford vehicle than a $3,000 Honda or Toyota. You’re less likely to end up with an overpriced problem child that way. My favorites are probably the large Buicks (LeSabre/Park Avenue). They drive very nice and have a lot of room for passengers and cargo. They also have one of the best engines ever made (3800) in terms of low maintenance and repair costs, good power, and good fuel economy. This engine was also used in the Pontiac Grand Prix and Bonneville, if you want something sportier and can find one that hasn’t been beaten into the ground by some teenage boy. If you want a small car, the Ford Escort/ZX2 are quite good, cheap, and easy to find. I also feel inclined to recommend the Chevy Cavalier/Pontiac Sunfire due to low resale value and high production numbers (read: there were tons of them made, so it’s easier to find a nice one). A lot of people absolutely hate these cars, and I may catch some flak on this site for recommending it, but I have not seen much bad out of them, mechanically (save for the Quad Four. Get the 2.2L instead). I have seen people get 300,000 trouble free miles out of them.