>>923800AZfag here, AZ is fucking god-tier for most /out/ related activities. I lived in Phoenix for most of my life, which during summer sucks-ass to be outside for any length of time, but during the winter time its actually very nice to go on hikes and camping.
Phoenix is right in the middle of the Sonoran Desert, which is actually pretty damn green, but it is still desert, and hot as fuck. However, a majority of Arizona doesn't look like that. A shit-ton of it is plains, chapparal, and in the mountains there are pretty amazing coniferous forests. It's a very diverse state.
Currently, I'm living in Flagstaff, which is only a two hour drive from Phoenix and is one of the snowiest cities in the entire continental US, and gets an average of 100.3 inches of snow each year. Camping opportunities are literally everywhere up here, me and my buddies like to just pick random forest roads and drive down them and pick random spots to camp. As long as you are on public land and stay under 14 days this is perfectly legal.
The Mogollon Rim is about an hour-and-a-half away from Phoenix and is a Pine forest with many lakes that is a very popular camping destination. My personal favorite place to go /out/ in AZ is in the White Mountains which is covered in Alpine forest, although it's about a 4 hour drive from Phoenix so that kinda sucks.
Fishing is pretty good if you know where to go, might have to do some 4x4ing to get to really good spots that aren't constantly fished out.
As for hunting, I don't do too much of that so I can't be too much of help. I've heard elk tags are really hard to get, but mule deer comes a lot easier. There are Coues deer populations in the mountains around Tucson (about 2 hours from Phoenix) and those are supposedly very easy to get tags for each year. Small game is literally everywhere.
Overall AZ is a fantastic state to be outdoors in, and I feel it's really overlooked. (Pic-Related, White Mountains this last March from a camping trip)