Terrific idea to hike the JMT from Mammoth area to Yosemite Valley, given your stated priorities.
Some observations to add about that:
Consider driving to Yosemite Valley for a day, seeing the sights (waterfalls, etc.), then leaving your car at the Happy Isles/Curry Village parking lot (standard parking location for Half Dome/JMT exits). Then go to the YARTS bus stop (will require a short walk or valley shuttle ride) and ride it to Mammoth. Spend the night in a Mammoth motel, or take the shuttle from Mammoth to one of the campgrounds in the Reds Meadow valley. There are several including Reds Meadow, Upper Soda Springs, Agnew Meadows (though I'm not sure the latter has been re-opened following the 2011 wind storm), and others.
Hey, if you buy Tom Harrison's Mammoth High Country trail map right now, you'll be able to follow this discussion more easily :)
Next, I fully agree that the JMT immediately out of Reds Meadow is worth skipping (I would even call it tedious). Think twice about doing the side visit to Devils Postpile at all...Better to spend that half-day (if you include the shuttle trips involved) on a side visit to somewhere far more spectacular right off the JMT, such as the legendary Lake Ediza (google-image that one) or Davis Lakes or Lyell Glacier. Anyway, start your backpack trip at Agnew Meadows for sure, and follow the JMT via Shadow/Garnet/Thousand Island Lakes. Far more memorable than the two parallel trails (River Trail or PCT).
I also agree with what someone here said about how the JMT from Tuolumne to Yosemite Valley is not nearly as interesting as the Agnew Meadows - Tuolumne section, but that if Half Dome is in your sights it is totally worth doing it this way, rather than attempting the very difficult Half Dome approach (and the permit lottery!) attendant to starting from Yosemite Valley.
Skip the side trip to the giant Sequoia trees. JMO. I know I may get flak for this as they are not the same as coastal Redwoods you saw last year, but...
While Kings Canyon & Sequoia Nat'l Parks are certainly spectacular from their westside entrances, I would say don't drive down there (westside). The drive there is not very pleasant, and you are already coming so far from Indiana. The eastside Sierra (the whole Highway 395 corridor) is absolutely spectacular and wonderful by contrast. I could spend weeks there! North of Lee Vining, in June, Bodie the ghost town would be terrific for a kid, as would Lundy Canyon, for the wildflowers. South of Lee Vining, there are endless varieties of dayhikes or backpacking trips in the Sierra eastside trailheads.
If your son liked backpacking the JMT and wants to do another BACKPACKING trip, I second the recommendation of taking North Fork Big Pine Creek trail up towards the Palisades. (Trailhead is west of town of Big Pine.) But the landscape there is so extraordinary above Third Lake, so I could never do this trail as a day-hike, I'd do it as a minimum 3-day hike so you can get all the way up to see Palisade Glacier. Sam Mack Meadow, on the way to the glacier, is, to me, the most breathtaking meadow in all the Sierra. Not realistic to get up there on a day-hike all the way from your car, unless you are in superb shape and acclimitized. Permitting via Inyo National Forest (recreation.gov) - it gets snapped up fast.
If you want to try just a dayhike, as a warmup or warmdown to the JMT, consider one of the following: (1) Onion Valley to Kearsarge Pass; take a gander of the breathtaking view from there into Kings Canyon Nat'l Park; return. This is some serious elevation training (pass is at 12k). (2) From Bishop, drive west to either Sabrina Lake or South Lake trailhead and just wander uphill, anywhere, for a day. Spectacular lakes everywhere! (3) Little Lakes Valley (Rock Creek) off the 395 not far south of Mammoth area is another good place for acclimitizing: trailhead is at 10k, although hiking is more flat/more of a stroll. I list these three hikes in downward order of difficulty (and also in south-to-north order).
Note that none of those 4 recommendations fall within any of the national parks. The entirety of the High Sierra crest/eastside trailhead system is identically spectacular whether it happens to fall within a national park, or not.
The 395 is a popular scenic route. There should be a good "highway 395 recreation map" available online that could help with your planning.
In Utah, Zion is probably my favorite of the ones you mention - you could spend many days there. Needles District of Canyonlands is my second-favorite, just the most fantastic fairyland ever, but that could actually be out of your way depending on which driving route you are taking through Moab/Arches. Arches is smaller, and astoundingly varied for its size, and terrific for a day or two of day hikes. Bryce is comparatively tiny, and personally I don't find it as compelling as those first three, but it may be "on your way" anyway, and is 100% worth stopping at to do one of the hiking loops for a couple of hours. (The nice thing about Bryce is that it's at 9000 feet and it may not be as hot as the other places in June!) The fun slot canyons in Escalante (as well as some much lesser-known ones north in Capitol Reef or east towards Canyonlands) are ideal for kids! Same principal applies in Utah as in the Sierra: Some of the very best hikes are not within a national park boundary. If I was taking a kid through Utah I would put Escalante on the top of my list and arrange everything else around that. Ask Dave C. here for even more detailed Utah recommendations, if he is willing to give them, as he seems to know a ton about all of southern Utah.
A little reluctant to recommend the long haul down to Death Valley & Grand Canyon in such a warm month, with so much to occupy you just in the High Sierra and in Utah. And most of the driving hours you would be adding are sort of tedious. But those two parks would be an excellent double-header for Spring Break sometime - fly into Las Vegas and rent a car.