There are several route options to the Grand Canyon. From Flagstaff you can go either north on State Route 89 to Cameron where you really should take a break and enjoy Cameron's Trading Post. Then turn west on Arizona State Rout 64 to the East Entrance of the Grand Canyon.
The road is well maintained inside the park. While some of the route is better than others none of it was any source for complaint. There are several picnic spots along the road and the scenery is breath taking.
Another option is to go west on I-40 to the Williams exit and head south on Arizona State Route 64 passing through Valle and Tusycan and into the south entrance. The Williams to the Grand Canyon information on Arizona Highway Guides has great information.
Even in the hottest part of summer it gets cool at this elevation after the sun goes down so bring your jacket and cool weather gear in case you're out late and of course experienced riders never head out without their rain gear.