You didn't say where you're staying for the Grand Canyon South Rim but if it's in the park you can take the park road 64 east to Hwy 89 and continue north. If you're staying in Williams on I-40 take I-40 east to Flagstaff and then pick up Hwy 89 north.
Hwy 89 north to Bitter Springs is an easy drive. At that point you can take 89A to Fredonia, AZ or 89 to Kanab, UT and then south to Fredonia, AZ.
If taking 89A you'll have a climb about 10 mi with curves 5-6% grade up to Jacob Lake (no lake by the way). From Jacob Lake to Fredonia you'll have a downhill of 6 miles of 6% grade. This route is done often by RVers as the area is a prime summer spot because it's cool.
If taking 89 from Bitter Springs through Page and on to Kanab, UT then Fredonia, AZ you'll have no issues northbound. It's a little different southbound.
Once you're at Fredonia, AZ then take AZ389 which becomes UT59 to Hurricane, UT. This is a beautiful easy drive. Then east on Hwy 9 to Springdale - another easy drive. Staying in Zion at Watchman Campground is the best place. Easy walk to the free shuttle and beautiful scenery.
Again, all of these routes are traveled by RVers.
If you happen to want to go over to Bryce Nat'l Park from Zion, the best way is backtracking on Hwy 9 to I-15 north and taking Hwy 20 east to Hwy 89 south to Hwy 12 east to Bryce. Don't take Hwy 14 off I-15. You won't like it!!