Are you planning a 'South Rim' Grand Canyon, or a 'North Rim' Grand Canyon? Or have you not decided? It will make a HUGE difference as to how you plan your travel.
Based on my fairly extensive family driving vacations with dogs around the Southwest, here's how to get as far as Denver:
San Diego to Vegas is the longest distance you should plan to drive! NEVER expect to drive more than 350-400 miles in a day except in highly populated areas. For every 2 hours in the car, you have to plan to stop for 20-30 minutes for children and dogs to pee and walk around--plus you have to plan at least 90 minutes for every meal break. And this 350-400 mile plan doesn't include any side stops to see any cool stuff along the way.
Vegas to Grand Canyon--complicated issue, because once you get to GC, you're basically surrounded by wilderness, and there are NO really good roads going east thru the reservations--seriously, N-O-N-E, NONE. If you're looking for EASY travel, then come back from GC to Vegas and spend the night again; then take I-15 north to Salt Lake City; spend the night in SLC, then go east on I-80 to Denver.
Most mapping software will tell you to turn east on I-70 to get to Denver from Las Vegas--but what the mapping software won't tell you is that I-70 is an EXTREMELY isolated road, with NO SERVICES AT ALL for 200+ miles--and no good services for nearly 500, from Cedar City, UT to Grand Junction, CO! If you break down on that road with kids or dogs, you could be there for 4-5 hours before the highway patrol gets to you, and then you'll still have to wait for whatever help they're trying to get to you; cell service on 70 is very spotty; not to mention that it's the primary drug smuggling corridor from SoCal to Denver. My cousin's a Utah Highway patrolman and his solemn advice to everyone who asks is "Never drive on I-70 west of Green River!" I've done it one time, and I'll NEVER do it again.
You're far better off driving all the way north to SLC, staying in a Motel Six, and taking I-80 across to I-35, and then I-35 down to Denver, with a stop in Laramie or Rock Springs. More people, more traffic, better services and emergency coverage. And your cell phone will work all the time.
BUT IF YOU'RE DELIBERATELY STOPPING AT THE GC to see it and can afford an extra day or two, then take what may be your last chance (since you're moving so far away!) to do the southwest tourist drive.
Here's what I'd do, kids, dogs and all, if I were you. See the GC South Rim EARLY in the morning, and than plan to drive from GC to Moab, Utah. Take highway 160 thru the Navajo Reservation, it's got the best coverage from the tribal police in case of an emergency. These roads are almost all 2 lane highway, but they're all paved and in very good condition, even though there isn't much in the way of services on the res; but once you hit Blanding, you've got services about every 40 miles or so and plenty of ranch traffic if you should need help. Also plenty of tourist stops, like the 'Hole in the Wall' and stuff like that.
You should plan an extra day/night in Moab if you can spare the time--this is the perfect time of year to walk the trails at Arches National Park--it's NOT too hot, and dogs are welcome if you pack out the poo!
Google "Slickrock Campground" in Moab, and see if they have any cabins available; I know for certain THEY love dogs! If they're full, make a reservation at the Super 8 and let them know about the dogs in advance. I taught in Moab for a year, and it's still one of my favorite places in the whole Southwest. It's amazingly beautiful. It's way more isolated that I was happy living in, but as a traveling stop, I can't recommend it more highly.
Your next leg is Moab to Denver. Be sure to take what they call in Moab "the River Road" to Grand Junction--any local can give you directions. It's not that much longer of a drive time than driving up out of the canyon to I-70 to Grand Junction, and you will never regret driving up that road, right down in the bottom of the canyon, next to the Colorado River. You can get on I-70 just before Grand Junction for the rest of your trip to Denver, but don't miss driving the River Road!
Once you're in Denver, someone else is going to have to advise you about breaking up the legs--I've only driven east of Denver once.
