Day 1: Rainbow Rim Trail. Drink Beer on side of Rainbow Rim Trail at Locust Pt. The trail is only 18 miles one-way, but you could easily do an out-n-back in 3-6 hours. This is a ride that you should do with a camera and a flask. Want more mileage? Head to Point Sublime that is just inside the park's boundary.
Day 2: Arizona Trail from the North Rim/Kaibab to Jacob's Lake is a must. This is a shuttle ride if you want to do the entire route. Sweet singletrack, big meadows, and the East Rim might be better than what you see at Rainbow Rim Trail. While picking up your car in the park, go to the lodge and have some pyramid hefe and pizza. If you want more mileage, add the final part of the AZT to the Utah border.
I have the GPX track and maps if you need them.