Equipment
Cornelius kegs are sold new or reconditioned. Either way is fine - it just depends on your budget. Just make sure your used or reconditioned keg comes from a reputable company and will not need major repairs.
Tanks, if purchased online, are not delivered filled due to obvious hazards and federal law. You will need to find someplace that will either fill your tank or let you exchange it for a new one. In fact, some beer distributors or homebrew supply shops will allow you to exchange your empty tank for a full one (fees vary, so check before going).
The regulator attaches to your gas tank and monitors the pressure in your keg as well as the amount of gas left in your tank.
Conventional wisdom suggests that it's better to have liquid tubing with a smaller interior diameter (don't ask me why, I don't get it). However, it is suggested that approximately 5 to 7 feet of plastic tubing with an inside diameter of 3/16" is ideal. You will also need the appropriately-sized barb clamps.
Before you buy quick disconnects, determine with you need the ball lock type or the pin lock type. If you don't know the difference, ask the person or company you buy the keg from or take it to your local homebrew supply shop and ask the proprietor.
Anywhere a hose connects to something else, it's connected by some sort of a barb. Everywhere a hose connects to a barb, you'll need a hose clamp. Just make sure you have enough barbs and hose clamps for all necessary connections.
To dispense the beer, you're going to be using either a picnic- or cobra-style tap or a draft beer dispensing system. The cobra-style tap is the black, thumb-operated tap you've probably seen at every BBQ and keg party since high school. Your dispensing system might be a dedicated kegerator or a converted refrigerator. All of this depends on your budget.
Kegging