Peepers,
The canyons are all stunningly different in Utah. A couple of tips:
Wyoming can be the coldest place on the planet...literally sometimes. In January it is not unheard of for the section of the freeway you have routed to be closed due to high winds (80MPH across the freeway), blowing snow, and temps of -20 to -40. Add to those temps if the weather is clear the fact that you are driving and the wind chill can drop the temps coming in to your radiator and surrounding your vehicle to sub arctic temperatures of -80 and lower. You need to prepare for cold weather through there. Keep emergency blankets, flares, 72 hour kits for each family member, small shovel for digging out of snow, 2-way radio/CB/HAM, etc.
Don't be scared, just be prepared. With all luck you won't need any of it, but if there is a problem, being stuck on the side of the road out there for 12 hours can be life changing.
Now, for the good stuff...I would expect temps in Southern Utah to be in the 40's to low 50's during the day in January. This would be good weather. From Moab hit Canyonlands and DON'T miss Goblin Valley. It is often overlooked, but don't. Watch ''Galaxyquest'' with Tim Allen. The place where the little green aliens are and the home of the rock monster is goblin valley and it is like the Disneyland of red rock country. It gets windy so a relatively warm day can be cold. Plan on cold weather hats that cover the ears and gloves and decent but unrestricting coats to get the most enjoyment out of it. Not a lot of people there in January so you can easily have the whole place to yourself. Very nice.
There is also a narrows canyon near Goblin Valley, Little Wild Horse Canyon, where the walls go up a couple hundred feet and kids can stretch their arms out and touch the walls on either side. January is probably too cold to go far back in because there is a section that fills with water and has to be waded, but it is a great hike. if you don't mind stripping down and drying off on the other side this could be a good thing.

They say that in some places the light is so filtered in the canyon that you can see stars in the day time. Definitely time well spent.
There is a dirt road that goes NorthWest from Goblin Valley to I-70 in the direction you are headed to get to Zion's and St. George. It goes through a reservation but provides some cool scenery and adds only a little time to the overall drive. It is very remote, meaning that sections aren't frequented even in hunting season, so if the roads are wet or snowy, avoid it.
If you head South from Goblin Valley on State Hwy 24 you can hit Capitol Reef State Park and the Water Pocket fold. Again, great 4WD trails and incredible scenery. It is not true that if you have seen one rock outcropping you have seen them all. If you continue on this route it'll take you past Bryce Canyon and Capitol Reef State Park. All of these are unique and offer incredible views and off road opportunities, as well as hikes that are suitable for the whole family.
Plan on spending time at Zion's National Park. You could easily spend the whole week there and not be disappointed.
The Coral Pink sand Dunes are a little South of Zion's. I've never been there personally, but I hear they are worth seeing. If you're road weary it's a good one to pass up.
Gas stops in Southern Utah can be few and far between. We live in Roy, UT and traveled to St. George several times last year. We planned our gas stop at the 1/8 of a tank point (Fillmore). When we got there a storm had knocked out the power for 150 miles in every direction and all the filling stations were dead. The nearest was 40 miles south and there was no word on whether they had power or not, and it would have been a real stretch of the MPG to get there. We ended up having to stay in an expensive cheap hotel with no power over night. It was 110 degrees in the room so the door had to stay open while everybody else who was also stuck kept their doors open and several partied outside the rooms. Not a pleasant experience. Take an extra gas can and stop frequently for fuel. We used the LED on the cell phone for a flashlight. Take one of those too.
Plan your gas stops in advance and don't expect every exit or named place on the map to have a gas station.
If you need to stop in the Northern Utah area feel free to let me know. I'd be happy to help you out anywhere within a couple hundred miles if you are stranded. If you need a place to bunk down or park overnight I'm happy to help. No garage for the 80 though, sorry.
Also, I have some good pictures of the Whiterim trail in Moab, The Needles area, Elephant Hill, Goblin Valley, and I think I have some of Zion's. Let me know if you plan on doing any of these and I'd be happy to share what I know.
Have a good trip, and good luck!
-M-