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>fly-fishing
If you want to catch trout, go to the streams and lakes up on and below the Mogollon Rim anywhere between Payson and Alpine (and east and south of Alpine is good too). There is some good Reservation fishing up there. Out around and east of Tucson are some fun (though small) trout streams on the "sky islands". Oak Creek, Beaver Creek, the tributaries of the Grand Canyon, and of course Lee's Ferry can all produce good trout as well.
Please don't abuse these areas, as trout populations depend on natural reproduction in most of them (and not stocking). Trout habitat in Arizona is fragile. It's a desert state, and we don't have the same kind of plethora of opportunities you can find in the Rockies. Read local regulations, don't take more than your daily limit, and, if you're fishing catch-and-release, when possible, use a net to land fish, play the fish quickly (don't drag on the fight), de-barb your hooks, handle the fish gently, don't use those nasty jigs with a billion killer hooks on them, and don't use bait (the trout swallow bait and often fight with the hook lodged in their stomach, as opposed to being lodged in their jaw).
>shooting
Any public land, I suppose. Arizona has more than enough to go around
>camping
It was a revelation to me when I realized that you can camp pretty much anywhere on BLM land, not that you'd want to. Most BLM land is the flat, uninteresting sagebrush high desert.
>hiking
I would run out of space in this post if I-