With just a little online research, you should be able to find good (mostly paved) roads that follow the trail quite closely. I-80 across Nebraska to Ogallala, then US 26, works very well into Wyoming. (One of those places to see ruts is Guernsey SP in eastern Wyoming, along US 26.) Around South Pass is where the Oregon and Mormon Trails split; before that point is a visitor center devoted to the Mormon handcart companies that passed that way, two of which got snowed in due to a late start, and needed rescuing.
In Idaho, some places you may want to stop are the little Oregon Trail museum in Montpelier, the village of Chesterfield (open Memorial Day to Labor Day, though you can drive by it on a series of back roads other times of year), Massacre Rocks State Park (off I-86), and 3 Island Crossing State Park (off I-84). Idaho is very Oregon Trail-conscious!
Both Massacre Rocks & 3 Island Crossing SPs have campgrounds with electric and some water hookups. Not many sites in Massacre Rocks can take large rigs, but you can see pictures & descriptions of each campsite online, which is helpful in making reservations. Chesterfield has an RV park but it's not listed anywhere - we just noticed it when we were there last Memorial Day. Baker, OR also has at least one CG, and there's a nice KOA a little east/northeast of Montpelier, about 3 miles out of town. One of the best Oregon Trail campsites we've had is at Robidoux RV Park, a county-run park in Gering, next to Scottsbluff. Good view of the bluff itself from the CG.