Planning to travel in that area in the spring is difficult because you don't know how long the snow will last.
I've read that people are figuring out that Yellowstone is best after Labor Day and that it's getting more crowded then--record crowds for that time of year. It still won't be like the summertime crowds, but it won't be deserted, either. And it will be cold, especially at night.
We were there in September 2011, and they were closing the campgrounds in Yellowstone on the 24th, and closed entire loops even before that, so it was difficult to get a spot despite the fact that there are fewer people there than in the summer. However, the CGs in West Yellowstone were still open.
Also, watch the closing (or opening) date of the Going to the Sun Road in Glacier NP. We wanted to get there in 2011 but just didn't have enough time by going to the Tetons and Yellowstone after Labor Day. Depending on your travel style, it could be pretty tight getting to the Tetons, Yellowstone, and Glacier between Labor Day and the road closure.
However, if you were to approach from the north, you could do Idaho and Glacier before Labor Day. Surely they're nowhere near as crowded as Yellowstone. And you could schedule to get to Yellowstone after Labor Day, and then head to South Dakota.
Then again, this can all be done in the summer without worrying about road closures and the like--it will just be a lot more crowded. But hundreds of thousands of people do it every year.
