You picked a great time to go. Not as many tourists.
Try Echinos Geyser, in my opinion, the best in the park, 1/2 mile walk through the woods. I think its in the Norris Geyser Basin, ask any ranger. Lots of large elk in that areas too.
Wildlife best in Hayden Valley in May and early June(buffalo, moose, elk, maybe even a Grizzly, Barrow's Goldeneye, etc. Try La Hardy Rapids on the Yellowstone River, north of Fishing Bridge, Harlequin Duck nest there (they are beautiful) and you can also watch the large Yellowstone Cutthroat spawning there. Yellowstone Falls is a must!!! near Canyon.
Of course, Old Faithful area, from there walk to Castle and Spazmatic Geysers and Morning Glory Pool. Riverside Geyser is also nice in that area. To the north of Old Faithful area, look for the Grand Geyser, when it goes off, its twice as high as Old Faithful.
If you want to see Trumpeter swans, the Madison River near West Yellowstone is best. Osprey and Bald Eagle, Lake Yellowstone.
I worked in the park two summers and have been to most areas.
P.S. For fun go hot potting on the Gardner River north of Mammoth Hot Springs. Its fun to wade/sit in 30 degree water next to 120 degree water and mix them to be effective and comfortable. Only problem, clothes are required.
have fun,
Van
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