The best way to hike South Kaibab Trail, from my point of view, is from the bottom to the top, having first come down Bright Angel Trail. My reasons for this are found here. In short, South Kaibab Trail is steeper, higher, more exposed, and lacks water, when compared to Bright Angel Trail.
However, if you chose to tackle it, here is my trial guide, starting from the Colorado River and ending at the top.
The trip from Phantom Ranch/Bright Angel Campground is a pleasant, almost level stroll along Bright Angel Creek, then branching off to the Black Bridge. The National Park Service records this distance as .7 miles, but it seems much shorter to me.
After crossing the Colorado River on the Black Bridge, you go through a short tunnel (no flashlights needed) and then start a sharp ascent up to the Tonto Plateau. On Bright Angel Trail, you do the ascent over a 3.2 mile hike, but on South Kaibab Trail, it takes about 2 miles. A few steps past the Black Bridge, you pass the intersection with The River Trail. Past the tunnel, South Kaibab Trail is immediately steep and strenuous. In cool weather you will find yourself down to your bottom layers of clothes in a short time, and when it is warm, you will heat up quickly. As you reach the Tonto Plateau, the trail straightens, running through red volcanic appearing rubble, before reaching the Tip Off. The Tip Off has an emergency phone and pit toilets, but no water (there is no water available anywhere along the South Kaibab Trail except at the trailhead [except Winter months] and at Bright Angel Campground/Phantom Ranch. Also please be prepared to be exposed to the sun and the wind for your entire hike on South Kaibab). Just past the Tip Off is the connection with the Tonto Trail that connects with the Bright Angel Trail 4.6 miles to the West.
Continuing up the South Kaibab Trail from the Tip Off, the trail is relatively straight for a time, then ascends a dramatic series of switch backs (the zig- zags) characterized by a steep ascent and large step-ups. If you have not yet been sweating on the trail you will likely start here. Toward the top of the zig- zags is a silver metal sign designating that you are about half way to the top.
After another somewhat straight trail across a ridge, you arrive at the ominously named Skeleton Point–a distance of 4 miles from Phantom Ranch/Bright Angel Campground and still 3 miles from the rim. Skeleton Point has no toilets or emergency phone.
From Skeleton Point you continue up a relatively straight, moderate ascent for 1.5 miles to Cedar Ridge, which has composting toilets, but no emergency phone. From Cedar Ridge you begin the final ascent of 1.5 miles to the rim, passing on the way, Ooh-Aah Point (.9 miles from the rim), where many rim hikers congregate to view the canyon before heading back to the bus.
The ascent steepens as you reach the rim, capped by a shaded set of switchbacks that are icy in Winter months, before reaching the trailhead, and the bus stop that takes you back to your destination. Depending on your starting point, you must first take the Orange Bus Route to Yaki Point, before heading back to the Visitor’s Center, then transfer to the Blue Route to get back to the Rim Lodges, about a total 45 minute trip on average (all bus rides are free of charge.) Or, if you are particularly energetic, you can walk the rim route the 4.9 miles back to the Grand Canyon Village. Good hiking to you.