The South Kaibab Trail is generally preferred to the Bright Angel Trail: it is more direct, and thus shorter; and the trail itself is more exposed, leading to better views of the Canyon, particularly eastward.
The Bright Angel Trail is not as relatively exciting: the views are more limited, and the section along the Colorado River is not especially dynamic. However, the Bright Angel may make for an easier ascent because it features four water sources, whereas the South Kaibab Trail has none.
Reaching these trailheads is equally convenient: the South Kaibab is close to Village Marketplace and the Bright Angel is adjacent to Grand Canyon Village.
From the Colorado River, the only logical trail to the North Rim is the North Kaibab Trail. If you are planning to overnight, other routes become feasible: the Tanner, Grandview, and Bass Trails all descend from the South Rim and can be connected to the North Kaibab Trail via the Tonto and Escalante Trails (the latter only if you descend via the Tanner Trail).
Time of Year
There are two primary factors in determining the ideal season for a RRR hike or run: temperatures and snowpack. Hours of daylight and water availability are secondary factors. In short, Spring and Fall are the most ideal seasons; Summer and Winter are doable but less desirable.
South-to-North-to-South, or vice-versa
The experience is different depending on where you start from, particularly in when you reach the steepest sections (at the start and end, or in the middle) and for how long you must go between water sources. But it is recommended to disregard these differences and to choose the most convenient starting point given your travel plans. For most people this will be the South Rim, which is easily accessible from Flagstaff and Phoenix. The remote North Rim is less popular, though it is more convenient for those driving from points north (notably Las Vegas, Salt Lake, and Denver). Note: you may not have a choice about where you start: the North Rim is only open mid-May through mid-October, depending on weather and snow conditions.
The speed record for RRR was set in early-November 2007 by Dave Mackey of Boulder, Colo., in 6:59:57. Traveling at a more recreational pace, most runners finish in 10-14 hours. Hikers can expect a minimum of 14 hours, with 16-18 hours probably being the norm. Overnighters will want to budget 1.5-2.5 days, depending on your fitness level, the type of experience you want (i.e. intensity), and the backcountry campsites that are available when you apply for a permit. Speaking of…
According to official park rules, backcountry permits are required for any “overnight use,” which includes not only camping but also night hiking, though I’d be hard-pressed to believe that Park Service rangers are patrolling the trails at night looking for bandits. Therefore, if you plan to crisscross the canyon in a day, you do not need a permit. If you plan to overnight—and, officially, if you plan to hike in the dark—then you need to get a permit from the Backcountry Office (see External Links, below).
If you start from South Kaibab, try to park at the picnic area immediately east of the junction of Desert View Drive and Yaki Point Drive. If you start from Grandview Point, park near the Bright Angel Lodge. (If parking spots are sparse, you may want to leave your car where it is and take the park shuttle.) At the North Rim, you can park at the North Kaibab Trailhead or at the Visitors Center.
Camping along the RRR route is permitted in designated areas only. These include: Mather Campground (about 2 miles from Bright Angel TH and 3 miles from South Kaibab TH), Indian Gardens, Bright Angel, Cottonwood, and North Rim (about 1 mile from the North Kaibab TH).
A detailed topographical map is recommended—while the junctions are clearly marked, a mistake could be disastrous. National Geographic’s Trails Illustrated map (no. 207) is adequate. To reduce your pack weight by a few ounces, I recommend printing a color copy of your route area. (If the copy is not bleed-proof, store it in a waterproof bag, like an Alokosak.)