I went to Yellowstone for a couple week a few years ago. It was pretty much just a fishing trip (didn't do all the sight seeing stuff, so I can't help you out there). Anyway, I hiked up above Yellowstone Lake (its a long hike, and there are no roads) and fished the headwaters of the Yellowstone. That is about the most memorable fishing I've ever had. There are lots of cutthroats, and they're pretty good sized too. And not another person for miles around.
But that doesn't help you much for a family trip. I also had good results on the Gibbon not far from where it meets the Firehole to form the Madison. It's flies only there, and I did well fishing a Prince nymph under a bubble on my spinning rod. There are some good browns there. But that area is almost too road accessible, and the fish are educated.
If the lower Gibbon is too warm, you might do well in the Yellowstone below the lake, or the Gardiner River. Both are pretty good streams. Also, the Upper Gibbon (above Gibbon Falls) almost always remains cool through the summer, and has pretty decent fishing as well, although the fish are generally smaller.
You can also do well on Yellowstone Lake. The cutthroat population is down, but there are still enough to be worth fishing for. The best part is, they generally stay within casting range of the shore all summer long. I caught a few nice trout out of there as well.