Agree with others who feel this is an awful lot of miles for such a short time. That said, some CGs we have felt were decent (at worst) to very nice (at best) that would be along the two (so far) specific routes people have recommended:
Cheyenne KOA: nice & clean, some sites for bigger rigs, some tent sites, lots of "in-between-size" sites, most pull-through, full-hookup or electric-water. The grounds are almost entirely river rock, which is a bit of a workout to walk in, but isn't dusty or muddy. Road/RR noise not too bad. We have stayed here a couple times & would again.
Rock Springs - again the KOA. Basically a gravel parking lot with hookups, but facilities are clean and it's a fairly easy on-off from I-80. Road noise not too bad, and the redrock bluff behind the CG is scenic.
For the Tetons, you DO have reservations already, don't you?? If not, you will probably not want to pull in too late at night, as you won't likely find a site. Haven't camped at Gros Ventre or Colter Bay (both NPS CGs, mostly no hookups) but many do seem to like them. Colter Bay RV Park if you need hookups - by far the best choice for the Tetons, IMO. BUT you DO need reservations.
Three days for the Tetons isn't actually a bad timeframe to visit; however, since you're right next door, especially if you haven't been to Yellowstone, you should spend one of those days visiting YNP. Spend a full day, drive the south loop in YNP - covers Lake, Old Faithful, Hayden Valley (buffalo), Canyon (waterfalls), and a host of other thermal features, potential wildlife sightings, etc. The other 2 full days you can spend enjoying the Tetons, and take a "side trip" into the town of Jackson. If you are towing a trailer or driving a large motorhome, do NOT try to visit Jackson (the town) with that rig - use your tow vehicle for that.
2020 Toyota Tundra CrewMax 5.7L V8 w/ tow pkg, Equal-i-zer
2020 Lance 2375