You might have some issues with water quality that time of year with the spring run off. Some rivers like the Lamar will not really be fishable. Any tail waters or bigger waters will be smoking like the Green or Mircle mile on the North Platt, or the snake river tribs which would be closer then Yellowstone. But Yellowstone is so beautifull at that time it is worth the trip. Slough Creek is still a great river but it is no longer a secret it once was. It is still worth the hike though. In the park I would think the lower rivers like the Yellowstone and the Gibbon would treat you the best. but I think they open mid July so you might be too early. It really depends what the spring is like. Firehole might be fishing good too. It will be toward the end of it's season. It is the coolest river to fish though.
PMD, Flavs, yellow sallys, maybe some salmon flies