Since you are in Arizona, why not start at Lees Ferry. Here is a trip report from another forum member to give you some ideas (if you haven't fished there before). Lees Ferry Arizona
Late March, early April would also be a good time to hit the San Juan river, it will be a bit cold at night but warms up nicely during the daytime and there is camping right there on the river.
Another location I would definitely fit into the schedule is the Green River below Flaming Gorge. I like going there in late April/early May, the wx can be pretty unpredicable making camping tough but the fishing is awesome and the tuber hatch doesn't really start until Memorial Day weekend. There is no camping allowed on the first 7 miles of the river (called the A section), but there are several campsites in the area. The B section has several very nice campsites along the river and this is a nice choice after the tuber hatch as the people floating stay on the A section with the rapids, so even in the summer months the B section has more solitude.
You stated that you are in good physical shape, with that in mind another option (mid-July and later) would be to come into Wyoming and do a back packing trip into the Winds. There are an estimated 800 lakes in those mtns with some great fishing opportunities for mainly cutties and Goldens. If you do consider this option get yourself the North and South Wind River Range hiking and guide map by Earhtwalk Press. I'd also highly recommend getting a copy of Rich Osthoff's book "Fly-Fishing the Rocky Mountain Backcountry".
Best of luck with your trip and I'm looking forward to reading about your adventure.