We're talking about a three-week (ish) trip, with a focus on the parklands of Southern Utah.
We would begin by going northwest (through the Ozarks?), taking a full six to eight days to get to out there.
Then we'd spend an entire second week exploring places like Arches, Zion, Capitol Reef, Bryce and Monument Valley. During that time we could either establish a single base and do a series of hub-and-spoke trips or else change lodging a couple of times along the way, probably mixing in some moto-camping.
Then we'd head across the border into Arizona, stop at the Grand Canyon (preferably a seldom-visited part of it) and then proceed due east across Texas (maybe spending a night camping at Big Bend and later catching some live independent music acts in Austin) and Louisiana (maybe including a short side-trip into New Orleans) and finally back to Florida over the span of that third and final week.
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(The map above is interactive, not a screen cap.)
We'd strive to keep the daily average mileage relatively low -- it's about a 6,000-mile route, so at 21 days that would give us about 300 miles/day, with an occasional 400-mile or 450-mile day here and there (when the weather's nice and the road and traffic conditions are easy etc.) plus a couple of zero-mile "break days" as well.
We'd stick almost entirely to secondary highways and remain well off the beaten path as much as practical. We would maximize opportunities for unexpected diversions and be open to schedule changes when things caught our interest. We'd try to make lots of time for checking out museums, historical markers and exhibits and weird roadside attractions. We'd take brewery tours, hike, eat at unusual local restaurants (NO NATIONAL CHAINS) and allow serendipity to guide us on a largely improvisational kind of journey between flexible objectives.