You can put in below Flaming Gorge Damó nice canyon. But the crush of $$$-mad outfitters makes it impossible to rig there for a long trip. The first takeout ramp (above Red Canyon) is a good place to load gear & food.
There are few rapids and much scenic meandering and birdwatching between there and the Gates of Lodore (NPS permit required). In warm weather the flatwater reaches can be buggy.
Below Split Mountain Gorge is another long stretch of meanders, irrigation shunts, and rusty junk car bodies. Then more of the same, to Desolation and Grey Canyons (another BLM permit). Then more ag stuff with dams, to Green River, Utah. Then Labyrinth and Stillwater. Slow and gorgeous.
Best description of the entire stretch might be found in River by Colin Fletcher (A.A. Knopf, 1997). Could be a great long trip, if you can work out the logistical part (and the permits).