This year in May and June, we visited some new ones and old favorites of the NPs. A few thoughts from this and previous trips:
It was very nice, sunny but cool enough to need jackets early in the day or at night, at the south rim of Grand Canyon in mid-May.
Yosemite's waterfalls were spectacular this May due to heavy winter snow melting, but the park was also already very crowded.
Some NPs can only be visited in late June or later. Check when roads and campgrounds will even be open in parks like Crater Lake (which had snow piled higher than the visitor center in early June), Mt. Ranier and Glacier. Their seasons are mainly July and August. North Rim of the Grand Canyon is cooler than the South Rim in summer, and is open from about May 15 to October 15.
The Canadian Rockies NPs and Icefields Parkway are worth the long drive up. We were there mid-June and everything was open and already busy.
Plan to see Waterton NP in Canada when you go to Glacier - well worth it.
Rocky Mountain NP was great in late June with beautiful snow-capped mountains but pleasant temperatures.
Yellowstone and Grand Teton can be visited in late August and early September when families are back home for school, but note that most campgrounds begin to close mid-September.
Utah's NPs are wonderful and an be really hot by August. Bryce at a high elevation is cooler in the summer than Zion. Be aware that mid June to the end of September is the Monsoon season of rains and flash floods in parts of the Southwest. I agree that spring or fall work best for the Grand Circle/Four Corners.