Use CarFax.com. It costs a bit, but it has lots of good info on any car you are potentially going to buy. If you are buying from a dealer, they should provide the carfax free of charge. If from an individual, you can get the car VIN number and run it through carfax. CarFax will generally tell you if there have been accidents, thefts, odometer rollbacks, etc.. associated with the vehicle. It is a great source of information.
I like to stick with Toyota and Honda. I have a yard full of Toyota cars and trucks, and the dang things seem to run forever. The Corolla is a good small car with good gas mileage. It looks like a box, and rides like a go-cart (not really THAT bad) but it is very inexpensive and very very reliable. If your goal is no-frills driving from point A to point B as cheap and efficiently as you can, you can't go wrong with the Corolla. If you want to step it up a notch and look and ride a little nicer, and stick with good gas mileage, then the Honda Civic is a good choice. It probably runs longer and has even less problems (if that is possible) than the Corolla. It has a better suspension and is a better overall ride. It usually has a few more frills as well. You will pay more for the Honda, but you get what you pay for.
Use KellyBlueBook and Edmunds to get an idea of what a car you are looking at is 'worth'. I usually shoot for 'person to person' price on anything I buy, but it's great to get it at 'trade in' value. DON'T get caught up in the car buying deathtrap of 'I gotta buy this before it's gone' or 'this is the greatest deal in the world'. The deal of a lifetime comes around about once every seven days. Check it out, test drive it, and do your homework before you commit to anything.
Good Luck and happy hunting!!!