Well that depends on what you want to do. Arizona has a nice balance of opportunity and trophy hunts. If you are set on hunting bulls then many of the November rifle tags can be drawn every 3-4 years. In AZ because our winters are so mild those November hunts are pretty challenging, more like post rut October hunts in other states.
If you are interested in hunting cows then many of the November hunts can be drawn with regularity, maybe 2-3 years. If good weather is your thing then the October cow hunts are pretty good for that and often there is still a bit of bugling. Those early cow hunts have similar draw rates to the Nov. bull tags.
September archery hunts in AZ are amazing, really the gold standard of hunting rutting elk. Those hunts take a while to guarantee a draw but there is a reason people are willing to wait.
The draw system is a little different than most other western states. For each hunt 80% of the permits are issued to applicants via a random draw. The other 20% are issued to those applicants for that hunt with the most points. So the good news is that every year you have a chance to draw any hunt you apply for.
Here is a good link to how the draw works, it's important to understand this to maximize your opportunities here.
https://www.azgfd.com/hunting/draw/process/
In terms of areas most of the elk are distributed through a band of high country that follows the mogollon rim across the center of the state. There are a few different pockets of really well known herds I.e. units 1/3/27 on the east side of the state, unit 23 in the center, and 8/9/10 in the NW. However most of the units have good numbers of elk and AZ trophy quality tends to be a notch above what you see in other states, however coming from NV that may not be as true as other places. Here is a link to the AZGFD website which does a nice job of explaining the units and densities.
https://www.azgfd.com/Hunting/Units/
I think you will find AZ to be a very hunter friendly state where you can carve out opportunity to have the experiences you want. You can wait
For years and hunt the best elk rut in the highest trophy potential units in the west or you can hunt cows every single year. For deer if you archery hunt you can hunt 9 weeks every year including the rut on an OTC tag while applying for the strip or Kaibab or you can hunt the deserts with a rifle every October. Bear is OTC, lion is OTC, archery turkey is OTC, spring shotgun turkey can be drawn every 2-3 years.
Antelope is a tough draw, you might as well consider it once in a lifetime.
Buffalo, Rocky Mt and Desert sheep are once in a lifetime.
Javelina is a draw but most units can be drawn every year and the archery hunts overlap the archery deer hunt and rut so it makes for some neat mixed bag opportunity and days with lots of stalks.
Welcome to the Grand Canyon state. And if you are interested in a more AZ specific forum I would suggest joining CouesWhitetail.com. It's the unofficial board of all things AZ hunting.