The typical kid kart is sold by most of the major kart chassis manufacturers that make cadet or full size racing karts. They are commonly 25 or 28 mm steel frames bent to shape and welded. Manufactures include Top Kart, Birel, Praga, CRG, Tony Kart, Emmick, Margay etc. New, they cost anywhere from $2500-$3500 including the motor. Some do not include motors. But don’t get sticker shock. Used, they can be occasionally found as low as $700 but often hit rock bottom around $900 and will hold their value at that point. You can find them at local kart shops, ebay, craigslist, etc. Often you will find someone moving up to bigger karts and might have a whole package to help get you going. A good thing to do is visit your local track and ask around about who might be moving up.
The two most common kid kart motors are the Comer C51 and the Honda GX50 which are both pull start. Comers are 2 stroke and are the most common except for in CA where the Honda, a 4 stroke, is becoming the engine seen most. The Comer retails for about $1000 and Honda around $700. The newest motor on the seen is the Iame Bambino M1. It retails for $1595.
A small toolbox for your normal hand tools will get you started. Bring a socket set, open-ended wrenches, pliers, Allen wrenches, and screwdrivers.
The most expensive thing you'll need is transportation to the racetrack. Kid-karts show up shoved in the trunk of a car, back of a pick-up truck, inside a van, snowmobile trailer, or even in a purpose built enclosed trailer.