You're going to both during the best time of year.
I'd call Zion Adventures about a Narrows shuttle. They might run a one of if you're whiling to pay. Start eary, bring neoprene socks and a warm coat in a dry bag, and a headlamp.
Failing that, I'd do an out and back from Weeping Rock up around the head of Hidden Canyon out to the overlook of the Great White Throne. You won't see too many people beyond Echo Canyon.
If you have a bit of climbing experience do The Subway. Bring a farmer john wetsuit, neo socks, and 60 feet of rope. No harness needed for the strong and fearless. Hitch hiking isn't hard to do up to the upper trailhead. Get a permit as early as possible the day before.
Or, if you're adventurous, have money, and are not afraid of heights, hire Zion Adventure Co. to take you canyoneering. Call and ask if Tom Jones is available.
For the Grand Canyon, it depends on fitness. The best dayhike on the south rim is down New Hance and back up Grandview, but that's a tough 18 miles and a dayhike for marathon fitness or better only. Second best is to take the shuttle over to the South Kaibab, take it down to the Tonto trail, across to the Bright Angel, and back up the BA. Go out to Powell Point while you're there. I actually think this is cooler than going to the river and back.
Free car camping at both places:
Park in the Coal Creek wash TH on the way into Zion, between Virgin and Rockville.
Right as you're leaving Tusayan you'll see a forest road going west (right). Drive it for a few miles, pull off, camp.