I have to disagree with part of saying the Grand Canyon would be fine.  The north rim will be closed until May 30.  I understand that is even with the road being passable earlier.  
Hwy 120 over Tioga Pass is normally plowed by May for the folks on the east side of the Sierra who want it done by April 1 for trout season.  
Yellowstone can be very cold in early May and snow is not unusual even in June.
We've been to Rose's at Bryce in April when they had to plow about six spaces for us spring breakers and none of the campgrounds inside the park were open.
All that said, there are usually work arounds if you can't camp inside a park and all of those places are beautiful when snow covered.
I wouldn't let the weather stop me from a trip like yours but remember to plan other places to go and things to see.  Check early for weather predictions and wear warm clothes.