I did a cross-Canyon hike about 15 years ago. I parked my car at the North Rim trailhead, took a van to the South Rim, stayed overnight at Tusayan, hiked down to Phantom Ranch in one day, then one day back up to the North Rim.
Figure out what time of year you want to do this. In the summer, the bottom of the Canyon gets pretty hot; in the winter, the rims get cold. I chose September, and still got a bit of heat exhaustion near the end of the first day. I made it to a water stop at the river and cooled myself off.
Choose which direction you want to go, and which route. The North Rim is about a thousand feet higher than the South, so I had to climb that much higher than I descended. There was a reason, though. There are two hiking trails from the South Rim to the bottom The South Kaibab Trail has better views, but there’s no water available. The Bright Angel Trail is more sheltered, so the views aren’t as dramatic, but there are a few water stops available. I decided to hike down the South Kaibab to get the views (packed plenty of water). The North Kaibab Trail is the only route to the North Rim, and there are water stops.
The reservation at the bottom was tough to get. When I did it, they started accepting calls on the first of the month, one year in advance. I started caling when they opened, got through 30 minutes later, and got the last bed for the month (so they said). If you’re doing a guided trip, I suspect they’ll take care of that for you, but I’m not sure.
Don’t take anything for granted. When I planned my trip, I found a van service to take me around the Canyon. The National Park Service website said they ran shuttle buses to Tusayan, so I figured I was all set. When I got there, the shuttle bus had shut down for the season just a few days before. A ranger told me it was only two miles to Tusayan, so I decided to walk. Turned out it was 3 or 4 miles to the Park entrance, and then 2 miles. Not the end of the world, but considering what was still ahead of me, the extra miles were a little annoying.
I did this 15 years ago, so anything might have changed since then.