I've only camped at the Lodgepole campground, but I thought it was a pretty convenient location, not all that far from the Giant Forest and General Sherman tree. Not all that far from Crystal Cave if you plan to take a tour of that. It's also not too far from the Lakes Trail, which goes to a series of 4 lakes starting with Heather Lake. I don't know if that would be too much uphill hiking for them or not.
You'll pass Moro Rock when you enter the park, assuming you enter on Highway 198 from Visalia. So you could either see Moro Rock as your first sight in the park or your last sight on your way out. I think the view from the rock is better in the afternoons. There are 400 stone steps to get up to the top of Moro Rock, and the top of it can be scary to be at, even though there is a railing there: …staticflickr.com/1179/1031495473_80b079befe… So for those reasons you may want to consider whether or not you want to take your kids up to the top of the rock. I think Panoramic Point, near the John Muir Lodge, is a safer viewpoint with about the same views of the Sierra peaks.
The campgrounds around the Grant Grove area are in Kings Canyon National Park so they are closer to the park features such as of course the Grant Grove itself, and Panoramic Point, and closer to the road that goes down into Kings Canyon itself, if you wanted to go down there to see the canyon, the Kings river, the Boyden Cavern, Grizzly Falls, Roaring River Falls, Zumwalt Meadow, etc.
So if you want to cover all these areas I might suggest 2 nights at Lodgepole and 2 nights at one of the campgrounds in the Grant Grove area.