You might consider visiting El Morro since you'll be passing right by it.
El Morro NM
is one of the most fascinating places I've been and you could combine stopping there along with the Petrified Forest stop (an hours visit). It's a huge mesa rising out of the plain, with a small pool at the base with ancient carvings in the rock...even some from Spanish conquistadors. It's been used as a resting place for travelers for thousands of years. Hike the path around the mesa and up to the top to visit the Anasazi ruins. Very interesting place. Only takes about an hour for the visit and hiking the trail. Well worth it.
After that, if you take a right onto US191 it'll be a short drive to Canyon de Chelly NM
. A spectacular ancient cliff community. There's been people living here for 5,000 years. The Navajo still farm the river valley. There's a RV park here for an overnight stay.
Then, 8 miles before Flagstaff right on I-40 is Walnut Canyon NM
, a spectacular cliff village built on a huge rock 'island' surrounded by a river 200 feet below with many ancient houses you can walk inside. Amazing place. Takes about an hour to hike the ring trail. Not strenuous. Many RV parks in Flagstaff but one of the favorites is Black Barts
. They have a floor show at the restaurant nearly every night.
The train up to Grand Canyon is a good choice...as the end of June is one of the most crowded times for the GC and you won't have to drive up there fighting traffic. The train arrives at GC around 11:30AM and departs at 3:30PM. Plenty of time to hike the rim trail a bit east, then take the bus back to the Village and hike west a bit. The buses are free and roam all over the south rim. You'll be hot and tired of fighting the crowds when you get back to the train. But it's worth it.
When you leave Williams, and head west, consider stopping at Grand Canyon Caverns.
This is a huge dry cavern 220 feet below ground (elevator ride) and is quite interesting. There's a nice large RV park here as well for an overnight stay if necessary.
Have fun! Maybe you should extend your trip to three weeks?