How much time do you have?
Interstate 40 might be a good choice. We've found it to be a nice drive. You could work your way up there at a number of places. I'm more familar with the west.
Just on the west side of Albuquerque there is Petroglphys National Monument. In Gallup there is Red Rock State Park, before Holbrook is the Painted Desert & Petrified Forest National Park. In Winslow, AZ you can go stand on "the corner" made famous by the Eagles song. Before Flagstaff there is Walnut Canyon National Monument. If you have time you can swing up to the Grand Canyon. There are many places along this route where you will find some sections of the Old Route 66. My two favorites are Seligam, AZ and from Kingman, AZ take Route 66 to Oatman, AZ and then it hooks back up to I-40.