i own a 91 mazda 626 LX, manual transmission, and although i wouldn't consider it an FAST car, it is very quick, quicker than a lot of cars you wouldn't think.
in 91 they also made a 626 Turbo, if you can find one i've only heard good things about them. If you can't find one, it's easy to find a 626, and the only difference between the 626 and 626 turbo is the turbo and anything directly related, so you could buy a turbo kit and have it installed.
but i would assume you are talking about newer cars, and i guess your choices depend on your definition of cheap. i would consider the neon SRT-4 cheap, but that's because i have the money for one.
i say **** a car, buy a suzuki GSX-R 1000