I spent a couple of days in Yellowstone last week.
All of the roads around Yellowstone Lake were still closed but expected to open some time in May. The road through to Grand Tetons was closed too.
From the North entrance, you could get to Old Faithful and the Canyon to the South, or East as far as Cooke City. The roads were still closed beyond that to the South and East.
There was still a lot of snow at higher elevations, but very little around Mamoth Springs and in the Lamar Valley (mostly in shaded areas - very little snow on the valley floor at all)
It can be quite variable from what I read, with occasional snow falls up until mid-June. Temp was about 20F over night, to mid-50Fs during the day in the park.
If you are interested you can see a few pictures at this link
The bison were just starting to have young, there were bears out on the roads and I saw a couple of coyotes and wolves feeding on elk. The bull elk's horns were about 2 ft long and growing fast.