The first thing to consider when backpacking down to the bottom of the canyon (the most common destination being Bright Angel campground), is getting a permit.
Permits are reserved through the backcountry office and can be requested 4 months in advance. At the start of each month, they start filling requests for dates 4 months out (ie: right now they are filling requests for April of this year). Camping spots along the main trails for prime hiking months (especially spring and fall) go quick - best to have your reservation in early. See details on the backcountry office website (link below). If you don't get a permit, you can try for the limited number of walk-up permits by showing up at the backcountry office when it opens (8AM) the day BEFORE you want to start hiking - there is no guarantee of getting one, so best to have a permit in advance.
As far as what is down there...
Bright Angel campground is pretty nice as far as backpacker campgrounds go. The campsites are all spread along Bright Angel Creek with lots of nice vegetation and trees separating spots to keep it from feeling crowded. Each spot has a picnic table, ammo cans (to keep food away from rodents) and a metal pole to hang packs from. There is a drinking water tap and a bathroom with sinks and flush toilets. You are also just a short walk from Phantom Ranch which has cabins and a canteen. You can buy snacks and a few other items at the canteen which serves as a rec room during the day and a bar of sorts at night (wine and beer). The canteen also serves dinner and breakfast that are aimed mainly at people staying in the cabins or dorms, but backpackers can reserve them too (done through the Xantera website).
One neat item to get there is the yearly Phantom Ranch t-shirt. Each year they make a new shirt with a special design for that year that can only be bought at the bottom (sort of a badge of honor for GC hikers). Fun things to do at the bottom include lounging by the Colorado River, hiking up to the Tonto Plateau on the Clear Creek trail for a neat look down at Phantom Ranch from above or hiking a little ways up Bright Angel canyon through the narrows. I also highly recommend the ranger talks they have in the evenings.
Many people like to hike down South Kaibab trail, spend a night or two at the bottom and then hike half-way up the Bright Angel trail to spend a night at Indian Gardens campground before hiking out the next day. At Indian Gardens, each spot has a picnic table covered by a ramada, ammo cans and hanging poles. There is a drinking water tap. The bathrooms are outhouses, but pretty nice as far as those go. There is a small ranger station and they have started having occasional evening ranger talks here too.
Note that fires are not allowed at any of the Grand Canyon backcountry campsites, so if you plan on doing any cooking, you will need to bring a camp stove.
What to bring depends a lot on time of year. I would avoid backpacking into the Inner Canyon from Memorial Day through Labor Day as temps get well over the 110 degree mark at the bottom - miserable hiking and potentially dangerous. In winter, the top of the canyon can have snow and sub-freezing temps, so you will want winter gear and crampons. Spring and fall are the best times, but you will still want at least a waterproof jacket and light fleece. Likewise, tent and sleeping bag suggestions depend on time of year. Having a head-lamp is handy for getting around and fixing meals in the dark. If you have not done much backpacking, trekking poles (look like ski poles) can make a big difference and take a lot of strain off your joints.
Have fun! It is a great hike!