I've been many times.....I'd wait till July 15th if you want to fish Slough, the Y. Stone in the Canyon, or the Lamar...The Firehole, Gibbon, and Madison would be a better bet earlier in the year because they are often closed due to high water temps. Fishing around Buffalo Ford on the Y. Stone is a complete waste of time these days.(No Cutts).
A vist to Y. Stone requires a bit of attention to rainfall, stream closures, and especially ROAD CLOSURES.....Visit the park website and check out the road work schedule before you make any plans.
Its usually cheaper to fly to Bozeman but price it....I usually fly in, rent a vehicle, drive to Livingston, then down through Paradise Valley to the Gardiner Entrance, then to Roosevelt Lodge in the Park.Guess that Roosevelt is about 2-2.5 hours from Bozeman.....Its cheap and they have a passable restaurant and a bar with a nice front porch....Though it might be a bit rustic for some tastes. You will probably want some lodging on the other side of the park if you want to check out the thermal features.......Make your reservations now....it will fill quickly.
There are two good guide books available...One by Craig Matthews, the other by Richard Parks. I'd buy both again.
Shoot me a PM when your plans firm up, and I'll share some favorite spots. Cheers.