Most people have been focusing their efforts on the Firehole River and Madison in the park. PMDs have been coming off in good numbers there during the day. And caddis have been hatching in the evening.
Most of the lakes are fishing well. Leeches and chironomids should be working pretty much everywhere. Look for calibaetis mayflies to be hatching on the lower elevation lakes.
The small streams are fishing well with attractor dry-flies. If you can't get them to come up, drop a beadhead nymph below your dry fly.
The northeast corner is still high and dirty. Slough Creek should be shaping up within the next week or two.
As always when in the Park, bring your bear spray and be respectful of the thermal features and the "no tread" zones. Read your regulations and know what's up.
Make sure to read the Yellowstone Park Fishing Regulations before you fish.