I went there about 5 years ago in April. There was still loads of snow around just to the south in the Teton National Park where the elks were, but once you got into Yellowstone itself, it was clear and much warmer.
It was quite surreal to be walking around on the wooden ramps next to boiling water pools. Most of the tourists tend to keep fairly close to the Visitor centre, so the more distant geysers and pools were more peaceful.
Once you get further north, it's very wild & remote.
