You've gotten some great advice!
If one of you is age 62 or older, you are eligible for a $10-for-life senior pass that is good for free admission and half-price camping in all National Park, National Forest, and BLM campgrounds. This bargain quickly changed our camping calculus!
Some National Parks (Monuments, Recreation Areas, whatnot) don't take campground reservations. Others do and there may be a slight additional fee for this service. It is probably worth it if you are camping during the high season as the popular park campgrounds do fill up. If you google each park of interest, you can usually find the page with camping info.
As others have stated, you have to plan on camping without hook-ups on most federal lands, but the NPS campgrounds generally have sani-dump stations and water that can be transported in a jerry can to your trailer if you run low.
Some BLM and US Forest Service campgrounds are on bumpy dirt roads where I wouldn't take a trailer, but others are fine. Actually, unless it's posted to the countrary, you can normally camp for free just about anywhere in a National Forest or BLM land, but unless you are familiar with the local area, it may not look inviting for an Airstream, from the highway. I think your best bet for unfamiliar country is just to find the district ranger office on-line, and phone ahead to ask about conditions. (Also snow in the high country persists late in the spring.)
If you go through western Colorado you will be close enough to visit Moab, Utah, with nearby Arches and Canyonlands National Parks, and Deadhorse Point state park. A nice and inexpensive BLM campground near Deadhorse Point is called Horse Thief, and they seldom fill up.
Ditto for Coulter Bay CG in the Tetons, from what we understand. We camped there one night, and it is huge.
If you go through Green River, Utah, there is a nice state park along the river.
We like to drop the trailer for a few days at a time, and go exploring in the truck.
Have a wonderful trip!