Dit it way back in 2007 on a rented Harley.

Yuma Simon explained what's not to be missed, but on my side I'd add also Oatman in Arizona (nice Old West town that is quite touristy but pleasant for a short stop), that you find after the 66 becomes a twisty road on the Black Mountains.
If you are into history of our planet, stop also at fabulous and huge Meteor Crater right outside Winslow (a small town on the 66 where you can snap a picture beside the statue of the character in the famous song of the Eagles "Take it Easy"), then don't miss a small stop at the famous "Roy's Motel & Cafe" in the ghost town of Amboy, California on the road to Barstow: the gas station, at least in 2007, was not operating anymore and was working only as T-Shirt and water sale, but they had some limited gasoline inside jerry cans that was veeeery expensive.
Consider that the stretch of the Route 66 around Amboy is much desolate and deserted (seen the movie "The Hitcher" with Rutger Hauer of 1986? It was filmed there!) and there was no gasoline available if not at Roy's for many miles (but consider it only as an emergency fill-up possibility).
The '50s look of the Roy's anyway, is unbeatable!


Follow Yuma Simon's suggestion to visit the Canyon de Chelly: it's not so known as the Grand Canyon, but it provides fantastic landscapes and it will not take so much time to visit.


UPDATE: the Roy's Cafe now sells gas again, see here Route 66 Roy's: History.

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Nick and his 2010 Yamaha XT1200Z Super Ténéré