Oh! ME, ME, ME!!!!
It's my favorite place in the world, and we've done quite a few backpacking trips. I'm super-jealous - I haven't been there in two years and I feel homesick!
Trips we've done - rim-to-rim-to-rim, starting on both sides. We prefer the South Kaibab trail - I think it's prettier, but because it doesn't have water, so you shouldn't hike up in any kind of heat.
We've also done some hiking on the Tonto Trail, combined to make a really big rim-to-rim-to-rim, again in both directions. Trailheads: Hermit's Rest, Grandview, and then the North Rim again. Oh - spend as much time on the North Rim as you can. SO much nicer than the South Rim - quiet, cooler, greener. The Tonto Trail is much more primitive, and water sources are harder to find in some locations. We had a scare one trip, and had to get our water out of a tiny little puddle with tadpoles. Yuck.
And, we've also hiked to Clear Creek from the South Rim.
When you get your permits and if you're planning on camping at the Bright Angel campground - at the Colorado River - try to book meals or even a cabin at Phantom Ranch.
Hmmm. Too much info???
For 3 days, I get to part of a thousand other journeys.