I recommend you look at this Audio Technica Bundle from LPGear. It includes a heavy, rugged turntable with aftermarket turntable mat and Audio Technica AT-95 cartridge, widely considered the best $50 cart out there. It has pushbutton controls to change speeds and even includes 78 rpm. It also has a built-in phono preamp and volume control so you can hook it directly to powered speakers if you like, or to a receiver that doesn't have a phono preamp. It also had an analog-to-digital converter and USB output so you can rip your vinyl to computer and on to your iPod.
And it's right at your price point.
And it's right at your price point.