Being so close we have been to Monument Valley several times and one of my favorite places to photograph. (See www.LaurenBarbara.com click the Wanderings link and the M V link under AZ). I also recommend Gouldings and make reservations for sure (ask for site 39).
We have done the tour only once - it is an open sided truck and you ride on benches / bench seats. Yes, you do get places not open to the "public" but we found these were not that much of value. Like anything else, depends on the "guide". And then stop at all the places that sell wares - some good, some just ok. But authentic to be sure. They also had a stop where about 5 young Indian girls danced - kinda just ok - but, of course, they wanted to be tipped.
My photo work gets me there very early in the morning and later into dusk. If you go to the website and link above you will see what I mean. (The very first image won awards - high acclaim on the other sunrise ones as well.
It has been $5 a head to get in and they now collect before you even get to the hotel they built a few years ago. But we could use that for both morning and afternoon shoots on the same day.
Adventure mention Valley of the Gods. This is north of Monument Valley a few miles. Well worth the trip!! And good road. Do not recommend the Moki Dugway (Hwy 261) going north from the west end of Valley of the Gods if you are not fearless mountain driver - and NOT, REPEAT NOT, for RV's. But beautiful.
Enjoy this wonderful area! Be safe.