Best of luck on your planned summer adventure.

I have camped many times in Yellowstone and the Tetons, always great places,
here are some thoughts:

I never make reservations just because I hate having to meet a schedule.  In
Yellowstone I prefer Norris CG which, as far as I know, is first-come-first
served.  Nice rolling/rocky area and an easy walk to Norris Basin geysers
making at least one day when you don't even have to drive the van.

If you'll be coming in the East Entrance you might camp in Cody or one of
the Forest Service CGs just outside the park to get to Norris by
mid-morning.

I have usually been there in the summer and though the big attractions (Old
Faithful & the Falls) draw crowds there are so many trails and less visited
spots that I never regard the throngs as much of a problem.

A couple of suggested family hikes:  Mt Washburn & Cascade Lake.

I always enjoy a swim in Yellowstone Lake -- usually cold but a bracing
experience.

East of Yellowstone in Cody WY is the Buffalo Bill museum.  I was skeptical
but totally impressed once I went.  Incredible collections of art and
artifacts relating to the American Indian and the West.

For the Grand Tetons: there is a Forest Service CG (Hatchet) just east of
the Moran entrance.  I generally stay there the night before entering the
park, makes it easy to arrive early at Colter Bay CG for a good spot.

An excellent Native American museum at the Colter Bay Visitors Center.

A float trip down the Snake River can be great for the family, these can be
booked at the Colter Bay shopping area.

Swimming is much warmer here, several nice spots on Jackson Lake that are
walking distance from the Colter CG, I like the north side of the peninsula
that juts out into the lake.

Family hikes in the Tetons:  Jenny Lake (hike, boat ride and a swim) & Swan
Lake/Hermitage Point.

Enjoy your trip & don't be like the guy I overheard at Canyon Village:
"Okay honey, we'll see the Falls right now then we can get to the 3:04
eruption of Old Faithful and still have time to make our 5:00 dinner
reservation in Jackson".

Geo