I’d love to move to western Montana or Idaho.
I don't blame you - Idaho is a great place to live. Except for 4 years when Uncle Sam made me live somewhere else, I've lived here my whole life, and I'm 72. The OP asked about a good state for big game hunting though, and I'll tell you right now - Idaho ain't it.
I’m really looking to go after elk moose deer and bears
Elk: Most of Idaho is "draw hunt only" for elk, meaning you have to draw a tag (and the odds aren't that good) unless you're talking about archery hunts. And the few areas that are "open elk hunts" (over the counter elk tags) are way over-populated with with elk
hunters during the season.
Deer: Only 1 in 3 Idaho deer hunters get a deer, and that's in a
good year. My wife and I do a little better than that in that one or the other of us usually gets a deer every year (I got one this year). However, my wife and I have some advantages - we live in the middle of some of the best deer hunting area in the state, we're retired and can spend as much time deer hunting as we want, and we have over a hundred years combined deer hunting experience. BTW - you're mostly talking
mule deer in Idaho. There's getting to be more whitetails around, but that's a fairly recent thing.
Moose: You have to draw a "once in a
lifetime" Idaho moose tag. Neither my wife or I have ever drawn one, and we've been putting in on them for over 30 years.
Bear: Idaho
does have some decent black bear hunting. I used to like the Spring black bear hunt up around Salmon. It's steep country, but beautiful.
Antelope (Pronghorns): You have to draw a tag. Neither my wife nor I have drawn one in the last 5 years. If you draw a tag though, your chances of killing an antelope are pretty good.
As far a big game
hunting goes, Idaho's one redeeming feature is that nearly two-thirds of the state is public land. That's 34.5
million acres, and less than 1% of that is inaccessible. So Idaho has plenty of areas to hunt big game without paying a lease or asking permission from a land owner. In other words, the problem with big game hunting in Idaho is
not the hunting, it's the finding.