Since Canyonag77 can handle NM quite well, I will just cover from Mesa Verde to Monument Valley to Grand Canyon. From Mesa V, you will probably go through Cortez and catch 491 south, then 160 to 4 Corners and Tec Nos Pos. Cortez has a number of restaurants along the main drag - I would definitely *not* recommend El Grande Cafe. The Ute Mountain Casino on 491 has a good restaurant, but I don't know about the ability to take kids there. Will probably be ok.
You have two choices to get to Monument Valley - the northern route to Bluff UT and then 163 to Mexican Hat, or the southern route through Tec Nos Pos/160/Kayenta. The northern route is much more scenic. Bluff has a nice little restaurant called Twin Rocks (the reason for the name is very apparent once you pull up), and Bluff Fort is a decent place to stop and stretch the legs and find a bathroom (cleanest in all of S.E. UT). Free admission there, and the Mormons who run the place are as nice as can be.
From Bluff you will pick up 163 to Mexican Hat. You will also pass the entrance road to Valley of the Gods. Research this before you go, as it is a great scenic drive. Just don't do it if it has rained recently. In Mexican Hat there is a decent little restaurant at the hotel/trading post situated right at the river crossing. Once you cross the river, you are now on the Navajo Reservation. Continue to Monument Valley, and consider stopping at the numerous pull-outs such as Forrest Gump Point. You will often find Navajo vendors selling their home-made items at these pull-outs. In Monument Valley, definitely stay at the View Hotel if possible. Goulding's hotel is historic as well, and has a restaurant and very large gift shop too.
From Monument Valley to Grand Canyon National Park, you will go through Kayenta and pick up 160 to Tuba City. Both cities have a few restaurants and fast-food places. If you want a restaurant in Tuba City, find the Hogan Restaurant on Main St., just past the Taco Bell. Don't go on Friday afternoon. Between Monument Valley and Kayenta, be on the lookout for volcanic remants, which are easy to spot as they are a different color than the surrounding landscape.
On 89 south to Cameron is the historic Cameron Trading Post. A bit of a tourist trap, but your kids will have fun roaming the place, and it's not a bad place to stop for a drink. From there, catch 64 to the east entrance to GCNP. The east entrance has a lot less traffic than the south entrance.
When you leave GCNP via the south entrance, you will probably go through Flagstaff and catch I-40 to Meteor Crater and Petrified Forest NP. In Winslow you will find the historic La Posada Hotel and their restaurant called the Turquoise Room. This is the best restaurant in that entire area. Holbrook has a fair number of restaurants, but the best one there we have found is called Bienvenidos, which although sounds like a Mexican restaurant, actually has a large variety. They do a very good job there, and is the only place we eat at in Holbrook.
From Holbrook you can catch 180 east to the south entrance to PFNP. Both entrances have visitors centers, but the south has more to see. From there you will make your way up to the north entrance, which sits right on I-40. There are a number of places to stop and gawk/hike, but we like the Blue Mesa area a lot. From there you can go east on I-40 to Gallup and the rest of New Mexico, which btw isn't new or Mexico. Gallup tip: don't eat at any Chinese restaurants there. The casino on the eastern side of town called Fire Rock does have a pretty decent restaurant.
A few more tips on AZ:
Four Corners is more of a tourist trap than anything else. Stop if it's a bucket list item, but otherwise I don't think it is worth it. However, the Navajo people there are quite friendly.
Don't drive at night on the Navajo Reservation. The Navajo do not keep their dogs or horses penned. Also don't let your gas get low as you drive around the place, as it can sometimes be difficult to find gas there.
Between Holbrook and PFNP you will see signs for a place called Knife City, which is right next to the interstate. Biggest knife shop I've ever seen, and the people there are friendly. Great selection, but most at list prices.
AZ does not follow daylight saving time, but the Navajo Rez does. Can't remember if the Hopi does too.
Lots of low-water crossings on the Navajo Rez. Be very careful if it is thunderstormy when you drive through. Don't leave you car unattended if yours is the only car around, particularly if you have non-AZ or NM plates.
hth,
C