You can get in some Route 66 aura by driving through the business route of I-40 in Tucumcarri and Santa Rosa. I would not try driving Central Ave in Albu the old route 66 route.
When you get to Grants and Gallup again take the business route for some 66 type sights and feel.
By turning south at Grants you can follow NM 53 to El Malpais and the Zuni reservation. From there you can go on over into AZ to US 191 or turn back up to Gallup on NM 602. I would go up 602 just to see the cliffs near Gallup and do some shopping in the old part of town.
Painted Desert and Petrified forest are worth a drive through and you can camp at Homilovi State Park near Winslow.
There is more old 66 driving in Holbrook and Winslow but the main old 66 route is west of Flagstaff at Ash Fork, to Seligman and over to Kingman and Oatman.