This route will take you right by the Petrified Forest National Park, and the Meteor Crater, both in Arizona. In California, you will be going through the Mojave Desert Preserve, which has some interesting points of interest. The town of Goffs on Rt. 66 between Needles and Barstow is nearly a ghost town, and has a museum about the old Mojave Road. Here's a link:
Goffs School - Mojave National Preserve (DesertUSA)
Here are a few more links to things in the southwestern desert (though not specific to I-40 or US 66):
Places to Go in the Desert Southwest - DesertUSA
Mojave National Preserve (DesertUSA)
Mitchell Caverns - Providence Mountains State Recreation Area (DesertUSA)
Ghost towns of the Mojave desert - DesertUSA
I know there is a lot to see in the southwest desert, and as indicated above, it's not just along Rt. 66. The links I provided should be good to get you started, but will probably also give you some ideas for vacations after your move. The best time to visit most of the desert areas is the spring, usually late March or early April, since this is the time when the wildflowers normally bloom, though this varies somewhat from year to year. One thing to remember, though, is not to venture out into the desert on unimproved roads unless you have a dependable four wheel drive vehicle and someone knows where you are going (and can get you rescued if you don't return). The desert can be very unforgiving if you end up getting stuck in it.
Anyway, enjoy your trip and everything you find along the way. It's an interesting and historical route, so you should find plenty of things along the way that are of interest to you.