I recommend three books for planning a trip like this:
"Eccentric America" by Jan Friedman
"Watch It Made in the USA" by Karen Axelrod and Bruce Brumberg
"Road Trip USA" by Jamie Jensen
The latter I use for general planning, and though it is about the old trunk highways, the Interstate system largely parallels those highways. It is just that the interesting stuff is off the Interstate, mostly along the old road.
First two were my late wife's choice for planning, she liked factory visits and the odd stuff. Jamie Jensen lists the oddities along his routes, but not the factory tours and company museums.
For the trip you propose, you could probably put together an interesting round trip loop using parts of Jamie Jensen's Oregon Trail, Loneliest Road, and Route 66 road trips (trips 8, 9 and 11 in the edition I am looking at).
You are making your north-south connection in California, so you will likely miss some of the red rock national parks and western Rockies parks in Utah and Colorado, normally visited moving between the Grand Canyon and Yellowstone. But if you want to learn about them, I recommend the Discovery Channel's Insight Guide "American Southwest."
For tent camping, I recommend finding public campgrounds in state parks, national parks, and national recreation areas. Across the whole of the Louisiana Purchase, you will find Corps of Engineers campgrounds on the flood control reservoirs for the Missouri, Platte, and Arkansas river systems that feed the Mississippi.
For private campgrounds, the KOA and Yogi Bear franchises require that operators provide camping facilities (sometimes air-conditioned cabins) on their property. Cabins cost more than a campsite, but much less than a decent motel room, and provide some comfort and protection when weather makes tenting miserable.
Not all RV parks permit camping, even some that include "campground" in their name. If you are using somebody's RV Park and Campground guide, check the descriptions to make sure tent sites are available.
Across much of the agricultural Midwest and Great Plains, municipalities and counties operate local low-cost facilities to support seasonal workers, but most of these I've stopped at have been strictly RV parks, no camping.
It is going to be hot going across the plains. My experience, that doesn't keep people from tent camping. They go to campgrounds where they camp in the shade (to have trees here means you are on a lake or reservoir), spend time in the water, and don't wear a whole lot of clothing. My campground outfit is sandals, light shorts, and a t-shirt.
But bring a little bit of warm clothing too, because it can get pretty cool at night and in the morning, when you get up around 7000 to 12,000 feet elevation,