I worked for the National Park Service in southern Utah for several years. In that job I was fortunate enough to travel all over that area. So for what it's worth here are my recommendations. BTW in offering these comments I am assuming you and your family are ok physically and can exert yourself on some fairly demanding walks/hikes. Also, assuming a 8-10 day trip.
What I am going to give you here is a route known as "the Grand Circle". It is world famous for those who love the western U S.
--upon arrival in Vegas drive to Zion NP. (About 3 hrs drive). Try to get accommations in the Park. Rooms are limited so you better make reservations now. If you do that you can take your car into the psrk and it is much more spectacular and convenient. If not, you can stay in Springdale just outside the park. Springdale is a very nice gateway community but not as nice as staying in the park. If you stay in Springdale you must take a shuttle bus into the park. Ok, but not the same as staying inside the park.
--I would stay in Zion 2 nights.There are 2 must dos.
1) Angels Landing..world famous hike. It's only about 3 miles round trip but it may well be the greatest short hike in the entire Park Service. It is a steep walk at some elevation so you will be sucking some wind but it will be unforgettable for you and your family. However if you are afraid of heights you may not want to do it. YouTube "Angels Landing, Zion" and decide.
2) Walk the Narrows. This walk is thru the Virgin River that runs thru the Park. In the summer it is a lazy river that goes thru a 2000 ft high slot canyon. The water temp should be ok if you there in July or later. Awesome experience. Again check it out on YouTube--"the Narrows, Zion"
--Drive to Bryce Canyon. (About 1 1/2 hrs drive.). Easy and scenic. 1 day there is enough. There are great cabins on the rim of the canyon. You have to make reservations now. If you don't get an in park cabin, you may want to just make day trip to Bryce from Zion and stay an extra night at Zion. Some great short and scenic hikes.
--Drive to Page Az. About a 4 or 5 hour drive with no stops. Very scenic drive including thru The Grand Staircase. Without getting too detailed there are many options here
-- Doing "the Wave". On the way to Page you will drive by this location. It is tuff to find by yourself. You have to plan this out and apply for a permit at the Bureau of Land Management office in Kanab Utah. They only allow a limited number of hikers pet day to enter. It's in an area called Coyote Bluffs. Do it. Your kids will,love you for it.
--Page Az. 2 days. 1 must do. Go to Antelope Canyon. It's on the Navajo reservation so you must go with Infian guides. Really unique and awesome. If you are into photos it is a treasure. Also, a boat trip on Glen Canyon / Lake Powell is really. Rainbow Bridge--a site near Lake Powell-- is worth it. It's a large natural stone arch in the wilderness off of the lake. Also, Horseshoe Curve on the Colorado River is an American icon. You will recognize it, when you see it. All these are within a few miles of Page.
--Drive to Grand Canyon. 5/6 hrs. 2 or 3 days. The must dos here are obvious. It all boils down to how much walking you want to do. Many short and long hikes. Of course if you want a lifetime memory walk to the bottom of the canyon and stay at Phantom Ranch, but you are probably too late to get a place there. In any event stay at the rim. El Tovar Lodge, hi$$$. Or Yosemite Lodge which is less expensive. Sky bridge is nice but I think it is $75 per person.
--Back to LV
