We just came back from the South Rim and had the best time bike riding there. We were very surprised that more people weren't doing it.
We brought our own bikes but parked by the Bright Angel Bicycles within the park (parking lot 4). This gives you easy access to the trails and your vehicle.
In order to get to Hopi Point for the mostly downhill ride to Hermits Rest, you need to put your bikes on the shuttle bus up to Hopi Point, the shuttles take about two at a time, so if there are several of you, you would have to divide up. You could also ride your bikes up to Hopi Point and onto Hermits Rest, however that would be strenuous. In our case, even though we had our own bikes we opted to use the Bright Angel Bike Rentals as our daughter is disabled and has her own bike. So, for that ride we rented bikes for ourselves and they took all of us and her bike to Hopi Point and met us about two hours later at Hermits Rest. Absolutely beautiful ride, the road is closed to cars so only the occasional shuttle bus on the very wide road, partway thru there is a bike path the goes off from the road. You get away from the crowds and see viewpoints most people don't see.
Back at the Bright Angel Bicycles, we used all our own bikes and set off for Yaki Point, a mostly level path that starts thru tree's (we saw elk here), and then follows a walking/bike path to Yaki Point, beautiful viewpoints along the way and very few people, (they are all at the visitors center).
After returning to the Bright Angel Bike Shop again, my daughter and I set off on the Greenway path to Tusayan, where we were staying at the BW. This path is slightly downhill, paved for about a third, and then hardpacked gravel and goes thru tree's. Very nice... My husband however, tried going from Tusayan up to the Visitors Center this way, very hard on the gravel and going uphill, so for an easy ride, going from the Bike Shop/Visitors Center is easiest.
Bright Angel Bike Shop was very helpful, and located perfectly for setting off on these trails. They have a great bike map and can point you in the right direction even if you don't rent bikes from them.
PS, if you have your own bikes and are heading to Zion, the Pa'rus Trail is absolutely stunning, goes over several bridges, and is very easy. And if you are also heading to Bryce the Red Canyon Trail is also great. On your way back (going West) from Bryce, get off about the summit, the trail follows the road, and coast downhill, and your driver (or husband in our case) picked us up near the end.