Contrary to popular belief, $2,000 can get you a lot of car if you do your homework. I don't think you can narrow your question down to 'best brand/model in range'. There are probably really good $2,000 cars in all brands and models, trick is to find them.
I probably would go with something domestic if you want 'cheap' (hehe), but seriously 'easy-to-fix', american popular cars that every mechanic is familair with.
$2,000 you are probably looking at something w/high miles (95,000+) - I would make sure the car you choose has had a lot of work done on it already, because at those miles you want to make sure that timing belts/chains have been replaced, struts suspension are good/replaced, exhaust and brakes are usually in need of replacement, etc. - just some of the things off the top of my hand.
-Vtrider