You will run into issues with snow in Teton, Yellowstone and Glacier in June. Some good hikes will be snowed out still. Teton is a hiking park, IMO. It's all about alpine hikes, high altitude and rugged rocks. September in the Tetons is incredible, but that is the only time I've been in them.
Glacier may not have the Going to the Sun road open in June. Very few hikes will be open or you'll be treading on lots of rice like snow. July isn't uncommon for the road to open.
Fly into Great Falls, Kalispell, or Helena. Hit the National Bison Range if you do Helena. The drive from Spokane is nice and a round of golf in Coer D' Alene isn't a bad thing if you like that sort of thing. There isn't a good way to get to Glacier from Spokane, so you've been warned.
Glacier, IMO, is also a hiking park. Hike the Garden Wall. Siyeh Bend, and Grinnell Glacier. You see a fraction of it from the road. Yellowstone is much more a road tour, although there are some great hikes. If you are a fisherman, you can catch east slope, west slope cutthroat and bull trout in Glacier. All in one day if you fish one side in the morning and the other in the evening. Yellowstone has great fishing too, but you won't see the west slope cutts. If you fish the lake, kill and eat every lake trout you catch. They aren't supposed to be there.
If you go to Glacier from Great Falls, take the Duck Lake road and stop at the Leaning Tree Cafe. It's a little out of the way, but a great place for breakfast and lunch. Dinner at Two Sisters on the east side is worthwhile.
In Yellowstone, get up early, like half hour before light, and get moving. You'll see more critters and have far, far, far less traffic. Take a nap mid day and avoid the idiots. This applies to all parks. GET UP EARLY. Spend a couple of evenings glassing bears on Dunraven Pass. You'll see huge mulies, elk, and grizzlies. That is the Grand Loop from Junction over Mt Washburn to Canyon. Go fishing.
Mt Al stole my thunder on the Cooke City Hwy over the Beartooth Plateau. It is worth it. Just be OK with switchbacks and being very close to the downcliff side of the road.
Jeremy