Yes, you can take your gun and ammo in checked baggage. There is plenty of discussion about that here on the boards. You will need a hard sided case for the gun and a lock that only you have the key/combo to for the case. Ammo in original factory boxes would be best. When you check your baggage, you tell the counter person that you have an unloaded pistol to declare. They will check that it is unloaded, you get a tag to fill out that goes in/on the pistol case. You lock the case up, put the case in your luggage, lock your luggage with a TSA approved lock. You will go directly to TSA screening where they will xray your bag, maybe have you open the case again to verify unloaded (I recommend just having the slide locked back) and off it goes.
The ammo can either be in the gun case or just in your luggage, but must be in a suitable container - the magazine is not considered suitable because it does not completely contain the rounds. That's why factory packaging is the best bet.
Once you get to Arizona, you can open carry, except in the standard prohibited places and I am not sure what those are in Arizona, you would have to check into that. I mean prohibited as in courthouses, maybe bars (not sure), schools, etc.
Opencarry.org - the Arizona forum....