For the deer portion I would start building NV for sure. You will have to buy a license each year so I'd put in for everything you're remotely interested in. With NV system you always have a chance.
Elk in most of those states will take longer I think, but there are some late hunts in AZ that a guy can draw in 6-7 yrs. I think. That said, IMO(for what it's worth) for the money-opportunity-quality for both deer and elk together.( Keep in mind out of the three criteria mentioned, that opportunity to go hunt ranks highest for me.) I'd rank NV first, then UT and AZ next out of the three.
UT takes a fair amount of points, but again you can put in for everything pretty cheap so to speak.(If you work it right you only spend 66$ every other year as it is now, plus app. fees).
I build points in all these states because it's too hard to choose for me though! In NV and UT I should draw within the next few years and in AZ I only put in for the lower tier late hunts, as I'd like to hunt there more than once in my lifetime

Good luck they all will be a great time!