Jason,
As has been said, GTNP is small compared to YNP. Nevertheless, there is a lot to see and photograph in GTNP. The grand landscape compositions are in your face, easy to understand, most are easy to access and close to your car. At this time of year, hiking will have to be at low altitude. In mid June we have found the GTNP trails, such as the Paintbrush Canyon and Cascade Canyon, blocked by snow, so much so that ordinary hiking boots were inadequate.
Just the same, I suggest you consider starting in GTNP. The weather is often better there than in YNP. You may not need reservations at that time of year; check that out. If so, a plan would be to spend time photographing GTNP for several days and then move on when ready.
You need to carefully consider where to stay to allow for sunrise and sunset shoots. Even in GTNP, the distances are such that if you are in the wrong place, a sunrise may be inconsistent with sleeping. We have found B&B's in Teton Village convenient, with more choice of dining than Jackson Hole or the more distant lodges in the park. We have also found that staying in West Yellowstone can be a convenient for YNP.