Have you considered the "A" and "B" sections of the Green below Flaming Gorge Reservoir? It's a beautiful canyon with huge fish and camping on the "B" section. Oh, and no permit required. The rapids are all minor IIs with one III-, Red Creek, that's a pretty straight-forward rock garden. Tons of wildlife, a gorgeous semi-wilderness setting and family-friendly water. The only downside, depending on the time of year, is a lot of drift boat traffic. And, the shuttle road from the end of the "B" section to Little Hole is pretty rough and steep as you climb out of the canyon.
You can never step into the same river; for new waters are always flowing on to you. - Heraclitus of Ephesus