Six days isn't much time.
Depends on time of year, which rim of the Grand Canyon you want to see and what you want to do. My recommendations are colored by my liking to hike.
My recommendations would be:
If you go South Rim, come in on I40 and catch Pertified Forest/Painted Desert and Meteor Crater on the way to the South Rim. You'll probably be getting to the Canyon pretty late on the second day if you push it. At the Canyon, be sure to go out the West Rim Drive to Hermit's Rest. On the way back leave by the east entrance, head up through Tuba to Monument Valley (remember that it is on the Navajo Reservation) and then up to Valley of the Gods. Drive up the Mogi Dugway and camp up on the rim. Or skip Monument Valley and head over to Chinle and Canyon de Chelly.
Alternatively, if you go South Rim, Flagstaff area has some interesting things to see. Northeast of Flag you have Sunset Crater and Wupatki national monuments. West of Flagstaff, there is a lava cave which you can hike through. There is also Red Mountain northwest of Flag on 180 (way to Canyon). Red Mountain is a short, relatively flat hike to a cider cone volcano that is u-shaped and has hoodoos. East of Flag is Walnut Canyon. To the south, you have the red rocks of Oak Creek and Sedona (plus a billion or so tourists) and the old mining town of Jerome.
If you go North Rim, weather is more iffy outside of summer and early fall. North Rim closes in winter. Some nice camp sites off of Fire Road 632 (hope the number is correct). From North Rim, you can hit Zion and Bryce national parks. If weather is permitting, you could take House Rock Road east of the Jacobs Lake off of 89A. House Rock Road is dirt and can flood out in spring. Nice hike in Buckskin Glutch (Paria River) which is the longest slot canyon in the southwest. Weather permitting, Cottonwood Canyon Road takes you from 89 up to Kodachrome Basin State Park and Bryce. I'm a mountain guy, not a canyon rat but Cottonwood Canyon Road takes you through some amazing scenery.