Plenty of good advice already. I will add a bit about weather. Summer can be very hot, so use a weather website like wunderground or weather.com to research average highs and lows for that month of the year at the various locations. My strategy in summer is to find higher elevations to beat the heat. At GC, for example, north rim is higher and cooler than south rim, and down in the canyon is hotter than hades during summer.
Your 32' TT will limit your campsite selections. In high-tourist areas like YNP and GC, you'll need reservations. However, as mentioned by another poster, some large CGs in Teton NP tend to not fill completely and this could serve as a home base for daily forays into YNP (although the drive times will be long, so this is an emergency fall-back strategy).
Once you decide roughly how many miles you want to drive daily (350? 400? Hopefully no more than 500), use the Woodall's online CG directory to see what CGs are in the vicinity of those approximate stopping points. Or use Allstays, or CampWhere app, or some of the other online resources. I like to have 2 or 3 alternate places in an area so I can adjust based on my progress (stopping early, running late, etc.) or if one CG happened to fill (move on to the next choice). If nothing is coming together, in a pinch you usually can park in a WalMart lot (ask inside as a courtesy)... although not sure if you'd want to put the slide out at WM.
Above all, for a great family experience, try to be always calm, relaxed, never anxious or uptight or in a huge hurry. (In other words, be the opposite of my dear old dad, LOL.) ;) Anything short of a severe accident is no big deal, you're just out having fun, chill out. Stressing visibly in front of the family can kill the fun really fast.
There have been a couple of other threads here in the past month or so about just such a trip including both GC and Yellowstone. Do a search and read what's already been posted; they had some good ideas about things to see and do in between those destinations.
