I have to add to the recommendations for the train from Williams. We got a package tour with the Auto Club (AAA) and normally I never take tours, but this did have some advantages. You stay the first night and the last night in a hotel in Williams right next to the train station, and they have a cowboy show right before the train leaves that is kid oriented. Then, on the train, they have live music and robbers that come through the train (they ride up on horses). Very campy for adults but fun and not scary for children. They give plenty of warning, so everyone knows it's a show, but I think it delighted the kids a lot. The food included with the package tour at the hotel in Williams was mediocre, I thought, but I think most kids would love it, and it was wholesome, just not gourmet -- standard cafeteria type food. We chose a package that allowed us to stay in one of the lodges at the canyon, and I would recommend that for at least one night. A bus tour of the canyon rim was included in the tour, but I would skip the lunch provided myself, although I think it would be fine for kids. It just was very mediocre "middle America" food I don't like. I would have bought something in the other part of the hotel and been happy to pay for it. Anyway, I thought the hotel was otherwise great. We had a nice room in one of the separate lodges that are scattered amidst the trees. There is a tram to take you from place to place and then you get out and can enjoy the walks along the canyon, the museums, displays, etc. There are ranger programs that include things for kids. If you stay one night at the Canyon, you have half a day the first day, and then until about 3 pm the following day. You arrive back at Williams that night, stay in the hotel, and then are on your way. You can go many other great places nearby. We went to one of the Mesa ruins to see the houses built into the cliffs. Sedona is near as well.
We did take our children many years ago and camped near the North Rim. Of course nothing was as well developed then, but even so, I'd favor the South Rim. The exhibits we saw were really great and enhanced what we saw walking along the Canyon rim. We did go early in the year and beat the crowds somewhat, although the train was fairly full and there were certainly people there. However, there were sections of the Rim where we felt as if we could enjoy the silence.
A highlight for us was also having a very nice dinner at a restaurant that literally overlooks the Canyon. We sat right by the window and could see the sunset reflecting on the Canyon. I have traveled a lot with children (I had seven) and even the smallest at age 5 still remembers, at almost 45, many of our experiences camping and traveling in Europe.