I've never been to Rocky Mountain National Park but have spent lots of time in both Yellowstone and Grand Teton National Parks. Both are great parks and only ten miles separate them.
If you want to camp in the Parks you have to stay at a designated camp site. You can go to the website, nps.gov and type in 'Yellowstone campgrounds' in the search bar for example and it will list all the campgrounds, maps, amenities and such. Same with the Tetons. Yellowstone has several campgrounds with hook ups but Grand Teton only has one, Coltler Bay campground.
Not sure of good boon-docking places near Yellowstone but there are several just outside of GTNP. On the east side of the Park, out past Kelly, you can camp for free on forest service land with views of the mountains, or out past the Slide Lake area in the Gros Ventre.
I don't know how much time you have, but if it's limited, I would make the drive over Dunraven Pass, see the Lower Falls on the Yellowstone River near the Canyon area and make the drive along the Yellowstone River between Canyon and Fishing Bridge. These are some of my favorite places in Yellowstone but you can't go wrong no matter where you visit. In the Tetons, there has been several grizzly sows and their cubs hanging around the Colter Bay area and the Willow Flats area all spring and summer. The 'inside' road within the Park has access to most of the lakes that sit just below the mountain range and the 'outside' road has beautiful views of all the peaks.
You can rent kayaks or bicyles at Dornans in Moose in GTNP and both Parks have good hiking trails. Local maps are available as you enter the Parks that will give you information on the latest events that may be going on.
Both Parks have great wildlife viewing possibilities and it's hard to beat the view of the Tetons rising out of the sage flats.
Hope this helps a little. You can PM me if you want more details.
Mike