I lived in Flagstaff, Arizona for 8 years and there are some beautiful roads throughout.
The primary highlights I suggest:
Take highway 89a from Jerome to Sedona to Flagstaff to Cameron and then cut over to the north side of the Grand Canyon. Jerome: a small, old mining town on the side of a mountain with a strong art community, many bikers (and tourists)... if you're thirsty and want live music, check out a bar called The Spirit Room. Sedona: famous, beautiful red rocks, spiritual / artistic hub of the Southwest. Flagstaff: the world's largest Ponderosa Pine forest, college town, population of 55k.
Note: Interstate I-40 also takes you between Sedona and Flagstaff, but you'll miss too much.
That trip on 89 will take you from that cool little mountain town (maybe 5,000 ft?) down into the desert-ish landscape of Cottonwood (a town that sits at the base of the mountain), into the red rocks of Sedona, through Oak Creek Canyon, up a couple thousand feet via the switchbacks into the forests of the greater-Flagstaff area, on toward Cameron through the open lands of a couple Native American reservations, and on to the Grandios Canyon.
Notes about Flagstaff: highway 89 takes you through the west and east sides of town via Historic Route 66. Also, to take a short detour, Flag sits (at an elevation of 7,000 ft.) at the base of a dormant volcano with a peak of 12,000 feet, a ski resort called Arizona Snowbowl, and twisty road to the top. The road starts about 10 minutes north of town off of Fort Valley Road and you can make it to the top (of the road, not the peak) in about another 20 minutes. But the ski lifts may be operable in the summer, so if you can ride up and then hike the rest of the way to the actual crest of the mountain, it offers a spectacular view. On a clear day, you can see the rim of the Grand Canyon. Also, all around Flagstaff there are Indian ruins (such as Wapatki National Monument, which also features ancient lava flows), meteor crater (one mile across... also, the astronauts from the Apollo missions trained out there because the landscape most-closely resembled the moon), plus, the road down to Wapatki takes you to the edge of the Painted Desert... fabulous!
There's quite a bit to see and do in the area, so that right there should be plenty to start with. Let me know if you want more information or have any questions.