What kind of camping rig will you have? How many kids, and what are their ages? That will help people to know what suggestions to make.
The long list of places Busskipper gave you were some places to think about and pick through; we all know that you don't have time for every single one of them.
It's fun to dream and plan, and you've got a good jump on that. Plenty of time to think things out and do research. I like to research places, too.
When going west for the first time, it's good to find out what the typical weather is like in the places you're thinking about at that time of year. I use a weather website like www.wunderground; I punch in a town/state and it brings up today's weather, then I scroll down and get the historical info and select the right month. That way I see typical highs and lows, record highs and lows, and precipitation info too. I don't know for sure what month you'll be traveling, but let me tell you, if it's between June 15 and Sept. 1 the temps will be mighty hot in desert areas below about 7000' elevation.
So you find out the weather info. Then you decide if you are going to feel ok in those temps. If not, start looking at a topo map to see where the higher elevations are at. Google Maps has a "terrain" feature you can select from the drop-down box in upper left of their screen, and it shows topography gridlines with elevation numbers. Figure that the higher you go, the cooler it will be. The north rim of Grand Canyon is about 8000' and is way less crowded than the south rim, which makes for good camping; search older threads here for info on places to dry camp in the national forest just outside the park boundary, for an "alone in the woods" type of experience that your kids might like.
If you want to drive only 350 miles per day, fine, it's your vacation and you shouldn't overdo it and make the family miserable. Get the map out and see how far 350 miles takes you each day. At each juncture, use online CG directories and info to select potential campsites near that point. If you don't like what you see CG-wise, you may have to adjust the day longer or shorter. But hopefully you'll see 2 or 3 (or more) acceptable places to camp near each end-of-day point. This is what I do... and I don't make reservations, because I prefer to keep it all flexible in case I change my mind and want to drive more or drive less or take a detour in another direction. But if you feel uncomfortable with that, then make your reservations; just remember that one sick kid one day can throw off the whole shebang.
Do you need reservations in the full hookup CG in Grand Canyon (south rim)? Almost certainly. And in Yosemite valley, you bet. But do you need reservations to see those places? No! The south rim's Desert View CG has no hookups but is a first-come CG, no reservations, and if you arrive at a decent time of day you should get a site easily enough. For the north rim, besides the boondocking I mentioned, there's a NF CG as well as Kaibab Camper Village (the latter is FHU) up the road at Jacob Lake. As for Yosemite, one can find first-come campsites outside of the valley but still in the park; plan to arrive there by early afternoon, Sunday through Wednesday, for best prospects. I spent a week at Yosemite... I snagged late reservations (1-2 weeks before departure) in the valley for two nights, and added 5 nights at first-come Bridalveil. My point here is, you can strategize and get around the need for reservations most of the time.
If you map out your stops on the way out and back, and add up the days you've used for travel, see how many days are left for the big destinations. If it's not enough time, adjust. Either drive more miles each day, or pick closer (or less) destinations. Above all, keep the trip relaxed and stress free as much as possible... or it's not a vacation after all.
There really is no place like Grand Canyon; that said, a few hours on the south rim was enough for the kids and me (back in the '90s). There's no place like Yosemite, either. Yosemite, though, is worth several days; I kept returning to Glacier Point day after day to take pictures in different lighting, eat ice cream from the little shop, and just sit and enjoy the view.
Grand Canyon, Bryce, Zion, Hoover Dam, and Las Vegas are all in fairly close proximity. From Vegas, it's one (long-ish) day to the Mono Lake area, from which one can reach Yosemite Valley in about 3 hours the next day (speed limits are necessarily low in the park).
By the way, if your kids are big enough, the Zion NP hike up the Narrows (walking in the river, with towering stone walls around you) is a do-not-miss, bucket-list-level activity.