x2 on Three Island Crossing. Wide grassy, shady sites with electric & water hookups, and there is a dump station. Historic area (Oregon Trail). I'd suggest staying in several different locations, as there is a different "flavor" in each one.
Two other good state parks along I-84 (or reachable from it) include Massacre Rocks, just west of American Falls (though not many sites available for large rigs) and Bruneau Dunes SP, south of Mountain Home, in sw Idaho - interesting large sand dunes in the park, and an astronomical telescope in the park as well; on weekend nights, volunteers come to open it to the public, and often bring their own smaller telescopes. Great & interesting program. You may need to make reservations; some of these parks can get filled up, especially on weekends. Idaho takes reservations for its state parks via ReserveAmerica, I believe up to 9 months in advance.
Craters of the Moon National Monument, on US 20, has a campground - very open, unique terrain, no shade, no hookups. Large rigs apparently will have some trouble negotiating the CG, though some sites will be large enough. Definitely dark skies there!
If you prefer private campgrounds, Mountain Home RV Park is an excellent place - exceptionally clean & nice. Along US 20, the Arco KOA is a nice CG as well, and fairly quiet - not right along the road (several blocks away) and not even near a RR track! :) Amenities include some shade, an evening ice cream social and morning (free) waffle breakfast.
Farther north, Ponderosa SP is a gorgeous gem. And VERY popular; you will need reservations there for sure. Also good advice about Yellowstone & the Tetons being close to s.e. Idaho.
