The typical Grand Canyon bus tour stops at the South Rim for up to three hours. That's plenty of time to explore, but only if you understand how to best use the National Park's free shuttle bus system.
There are four shuttle routes, but only two matter to tour bus travelers. Those routes are:
Blue (Village Route)
Green (Kaibab Trail Route)
(The other two - Red (Hermit Rest) and Tusayan - will consume too much of your sightseeing and gift-shop time.)
Blue Route
If you are strapped on time, stick with the Blue Route. Most tour buses will drop you off in front of Bright Angel Lodge, the main landmark in Grand Canyon Village.
As such, walk about 200 paces to your right and you will find a Blue Route bus stop. Shuttles run every 15 minutes from 6:30 a.m. to 7:30 p.m. local time. The bus' destination sign will read "Village Route" or "Blue Route." On board, the bus will head west for 100 yards and bend left/South en route to Maswick Lodge, before heading east to the new Grand Canyon Visitors Center.
I recommend that you stay on the bus until you reach the Visitor's Center, which is also host to the transfer area for the Green Route. Get off the bus, head towards the Visitor Center. Here's a good place to take a "comfort break" (the bathrooms are very clean). Follow the path and signs north for about 150 paces, cross a two-lane road, and enjoy Mather Point, one of the South Rim's most dramatic lookouts.
Time is of the essence, so I don't recommend taking the Rim Trail west to Yavapai Point, the other "must-see" Rim lookout. Return back the way you came to the shuttle bus station and board any Blue Route bus; they all are making the return back to Bright Angel/ Grand Canyon Village.
In about 10 minutes the bus driver will announce Yavapai Point. I highly recommend this view. It's got a great set of lookouts and a fascinating museum with attached gift shop. If it's summer, the north section of Yavapai Observation Station is all glass, giving you an air-conditioned, panaromic view of the Canyon.
Go back to the shuttle bus stop where you got off and return to Bright Angel Lodge and the main tour bus drop-off area.
Total round-trip time to ride the Blue Line non-stop is 60 minutes.
Green Route
Only consider taking the Green Route if you decide to spend the bulk of your time looking at the Canyon. If you include the Green Route, expect to reduce time spent in souvenir shops.
To get on a Green Route bus, you must ride the Blue bus to Mather Point, where the Blue Route ends, and transfer to a Green Route Bus. Green busses come every 15 minutes, so you will not have a long wait. Board the Green bus, which will take you further east. Stay on it until the driver announces Yaki Point, which marks the end of the Green line.
The view from Yaki Point is stunning and offers you eastward views of the Canyon that you will not see from any of the Blue Route lookouts. Yaki Point is also less congested with visitors. Many times it will be you and just a few others enjoying this outpost.
Go back to the Green shuttle stop and board. You are now heading southwest toward Mather Point/Bright Angel Lodge. The first stop will be South Kaibab. I suggest you skip it and get off at Pipe Creek Vista, from which you will get a great shot of the South Rim as well as the South Kaibab Trail and more. From here, take the shuttle back to the Visitor's Center, transfer to the Blue Line, and head back to Bright Angel Lodge and your tour bus.
Total round-trip time to ride the Green Line non-stop is 30 minutes.
Parting Words
To get the most of your South Rim bus tour, prioritize your time. How much time do you want to spend visiting lookouts? How much time eating? How much time visiting the numerous gift shops? Overlay that onto the amount of time you will spend in total at the South Rim. This will determine how many stops you will make on the Blue and Green Routes.