Reports from the locals are consistent that the Madison Valley streams, Madison River, Gibbon River and Firehole River, are in good shape for the early season. As almost always during the early season, the best opportunity is on the Firehole River. Hatches of small Blue-winged Olives, size 18 baetis species, are occurring in the afternoons. There's also reports of decent PMD hatches taking place on the Firehole.
The Gibbon is running much colder, of course. The mixture of its water along with the warmer water of the Firehole River has the Madison River in decent shape to fish. You would probably want to fish streamers and larger nymphs on the Madison.
Most all the other streams in the park are not in good shape, fishing wise. The Gallatin is clear but very cold. The Yellowstone is still high and cold. The Northeast corner of the park is probably out of the picture until well into July.
My guess is the streams will remain in good shape at least through Saturday and possibly through Sunday, depending on the amount of rain and melting snow. The warm weather and rain that's expected will bring about much higher water levels by the first of next week.
If you are can get to Yellowstone in a hurry, you will probably encounter some good fishing opportunities. Fishing conditions inside the park next week, aren't looking so good although I'm sure there will be places you can catch trout. There's a good snowpack and it will be melting at some point in the near future.