Good points above:
Check the az dnr for seasons and regulations and training requirements
Your airgun will be illegal for game birds most likely
I can think of several types of hunting:
Sitting quietly in nature at sun up or near sun down
Walking super quietly and listening as you stalk, stopping every 3 or 4 steps
Watching a pile of bait... not for me, but it’s a thing.
Using your low power air rifle, you will be fairly limited to cotton tail rabbits at close range with good shots to the head (between the eye and the ear). Even then, you are underpowered and that means you might wound an animal for it to only run away, suffer and die later. That’s not ethical hunting.
As said above, bows are cool and have a lot more power.
Buying new, you can get a ruger 10/22 for about $200 at large stores like WallyWorld. A Marlin Model 60 costs about $20 less and is just as good but in a different style with less of a fan base. There are some $120 rifles out there and some are great and some are junk, all with the same name on them... so you pay a little more for consistency and support if your new rifle is a clinker.
Sometimes you can find great deals under $150 on old bolt action rifles that are not ready to accept a scope. You like iron sights anyhow so this is to your advantage.
I think you can have a great time exploring the area near your house. Find a spot with a good view of cover and slink out there just before dawn with a little pad for your butt and just sit there perfectly still for an hour. You’ll see something. You don’t have to shoot it.
When you are ready to shoot stuff.. I have a side bag to carry a whole bunch of stuff like a hunting knife, nitril gloves, plastic zip loc bags, a mesh bag that used to hold oranges, paper towels, rags to clean hands and knife, matches, string, I am probably forgetting something.
Rabbits around me have had fleas and ticks so I didn’t want to just chuck them in my field jacket so I take out the guts and skin them and pop the meat in a plastic bag before going too far. I use the mesh bag to carry the meat in the zip locks outside my bag so the meat cools to slow down bacterial growth. I had tried hanging the carcass from a tree to go back for, but hawks and vultures were looking at my bunnies so I decided that wasn’t working for me.