Some of the state parks can be hard to get into, even on weekdays (Ponderosa comes to mind.) But they are worth it. Others have mentioned some of our favorites: Ponderosa and Farragut especially. We have also enjoyed Bruneau Dunes (local astronomy club puts on a presentation with telescopes on weekends), Three Island Crossing (lots of shade there too), Hells Gate, Massacre Rocks, and Henrys Lake (the last can be quite windy, but there's a 360-degree mountain view). A few of the state parks have full hookups, and most have both electric & water at many of their sites.
For FHU CGs we like Mountain Home RV Park and Village of Trees RV Park in Declo (between American Falls & Twin Falls, a good overnighter).
Some of the NFS CGs we have seen in Island Park (between Idaho Falls & Yellowstone) are really nice, and a few have some electric hookups.