Vegas has a lot of kids things, or at least that kids will enjoy. Aim for a hotel with a fun pool preferably center strip. I would spend two days, maybe 3 if you want a lot of pool time, in the city. Probably pick up the cars the second or third day and go to Freemont Street that evening.
See Hoover dam on the way out of LV going to the GC. Plan on 2-3 hours there. The Hard Hat tour is great, much better than the basic tour.
You can see some of the GC that night. Overall you can visit the south rim in a full day, but if you want to do some hikes into the canyon add a day for each hiking day.
From there it's about a solid days drive to Anaheim, but you might want to stop at a ghost town somewhere.