I'd recommend that you leave by May 1 as you will have a long drive to get out west and you'll probably want to make some siteseeing stops enroute.
Grand Canyon North Rim open May 15. Having done both Rims many times, the North Rim is now our favorite because of the cool temps and less crowds.
However, if this is your first time for the Canyon, I'd suggest you visit the South Rim which is open year-round so you could even get there earlier. There's more to do and more of the Canyon is see at the South Rim. Stay in the park at Trailer Village RV if you can. You'll be within walking distance to the Rim and also the free shuttle stops at the RV park and you can get on and off wherever you want. Definitely walk down the Bright Angel Trail if only for a short 1/8 mile or so. You'll get an altogether different perspective of the Canyon from below the Rim. There's also the easy Rim Trail to walk. Take the shuttle to your starting point and pick up the shuttle when you get tired. It's a very pleasant walk.
Then go to Zion (Watchman campground in the park has electric).
Bryce - dry camping or stay just outside the park at Ruby's or a couple miles on Hwy 12 at Red Canyon national forest campground - a beauty.
If you have the time then go via Hwy 12 to Capitol Reef Nat'l Park and then to the Moab area for Arches/Canyonlands Nat'l Parks.
Then head north for Grand Teton Nat'l Park - stay in the park at Colter Bay RV (full hookups) or next door at Colter campground (300 dry sites - no reservations) or just north of Jackson in the park is our favorite, Gros Ventre campground (again, 300 dry sites - no reservations). The two dry campgrounds rarely fill and you'll get a site. We've done so, many times, with our 40' motorhome.
Each national park has a great web site giving things to do, maps, campgrounds, road information and special park alerts. Check them out.
Have a great trip! All are favorite parks of ours!!
