Congrats on the hall pass from the wife for a Yellowstone trip! I grew up in PA but now live in Bozeman and we frequently help folks plan trips in the Yellowstone area. Early summer (late June or early July) is a great time to visit the greater Yellowstone region. This is a transition time of the year for waters in YNP and Montana and conditions will vary from year to year but you will be garaunteed some great fishing. I would avoid the Gardiner entrance and the waters in the north eastern section of the park like Slough, Soda Butte, Lamar, etc. if you are planning in advance for dates befoer July 15th. The risk is that these waters will still be too high depending on the snowpack (about a 50-50 chance). A safer home base for early summer will be the western section of the park. The catch is that the Firehole and Madison in the park warm quickly and on some years are too warm in early July. Grayling, Duck and Cougar are also great options in mid June (small streams near West yellowstone)
Another great fishery that is always good in June and July is the
Madison River outside of the Park. Hebgen dam helps moderate the flows and the sections above the West Fork are fishable regardless of snow melt. The Madison is a big river and some parts are best fished from a drift boat so I would skip trying to DIY between Lyons and Ennis and hire a guide for that day. The best DIY water is from Hebgen to Lyons bridge, also some good wading in the Channels below Ennis.
Some of the fisheries in the park can get a little crowded in the peak summer months. I would almost recommend the 3rd week of June when the Firehole is a sure bet and most likely the Madison in the Park and definitely the wade section of the Madison below Hebgen. I would also recommend booking a rod or two on the
spring creeks near Livingston like DePuy, Armstrong or Nelson for the pmd hatch which kicks off around June 15 and runs strong into early July. If you fish some of the more technical waters like the Letort you will feel right at home. The pmd hatch is a lot like the sulphur on Spring Creek or Penns but comes off mid day. Those rods book out far in advance FYI.
The nice thing about the third week of June is that the rivers will still be mostly empty and the wade sections will not be that pressured - they can get pretty busy in late June and early July.
Another idea is to push the trip back to mid July and target the great attractor dry fishing on the other end of the park and base your trip out of Paradise Valley on the
Yellowstone River. You could hit Slough, Soda Butte, Gardner, Lamar, etc. inside the park, do a float on the Yellowstone River outside the park and also catch the Boulder river (amazing attractor dry fishing in mid July). The spring creeks would also still be a fun option and they are right there to mix it up a bit. I would wait until July 15 to play it safe in case of a big water year if you are planning in advance.
Some other great fisheries outside of the Park to consider are the Ruby and Gallatin for DIY fishing - they are both smaller fisheries with a lot of public access and good for wading.
Feel free to drop a line if you need advice on where to go, especially if you are planning on a lot of DIY fishing. A lot of the rivers you read about are too tough to fish by just showing up (some too big for easy access) and others are just right. Having a bit of inside knowledge can go along way to make sure you have a great trip. Enjoy!