The thing is, simply taking up rappelling doesn't get you anywhere, and, as a stand-alone activity, I can't say it's particularly safe. Lots of mountaineering accidents happen during rappelsanchors pull out, people rappel off the end of their rope, descending gear jams. I've climbed for 25 years, and I rappel only when there's no other way to get down. The fact is, I plain don't like it.
Be that as it may, the basic equipment list goes like this. A rope, of course. For all-around use, a 10.5-millimeter-thick, 60-meter-long rope is best. Something like the Maxim Leavittator ($159), which also has a water-repellent coating so it doesn't get too soaked in wet weather or on snow. Next, you'll need a harness. Black Diamond's Momentum ($42) is an excellent all-around harness, comfortable and strong. Everyone has a preference when it comes to rappelling devices, but I've long used the Black Diamond ATC (Air Traffic Controller$17). This is what's called a "straight-through" rappel/belay device, as opposed to a figure-of-eight model that, while strong and simple, has been known to twist the rope as it passes through, therefore kinking it. You'll require leather gloves, of course. Any good-fitting pair from a hardware store will work; the fit is important, as you don't want any loose leather getting caught in your rappel device. And lastly, nylon slings (make your own from one-inch webbing material) for setting anchors.
Two words of advice. One, instruction. Do NOT try to figure this out on your own, set up a rappel station, and take off over the side of a building. Two, caution. Rappelling is not a game, nor is it recreation. It is a hazardous descent method that should be treated with respect. Properly executed, a rappel is efficient and safe. But the safety margin is exceedingly thin, and any single failure will cause the entire system to fail, with the obvious grave repercussions.