Let me see if I can help. Being from NY, and once in your shoes on wanting to go west for Elk. I chose Colorado. It has the most Elk of the western states. It seems like a natural choice for you being closest. Non-resident tags for Colorado are less expensive than most the other states now. $551 for a Bull Tag is plenty enough out of my wallet, too much actually. But its better than a Montana $1,000 plus tag!
Do It Yourself (DIY) hunt your in for alot of research. You have to pick a season type, archery, rifle, Muzzleloader likely won't be a option w/o any points to start. Then look at different areas, units, and tags avail.. Most archery tags are over the counter(OTC). 1st rifle is a draw only tag, 2nd rifle is OTC. So I'd start looking at that. When looking at units, look on maps, and see what kind of access you'll have. Some areas are wilderness accessible by foot, and horse only. So its not as simple as driving to a spot, and going in. You'll need to consider tenting it, or hoteling it. Hoteling it will most likely require daily trips to hunting grounds, and hiking from there. Tenting you can pack gear in, and setup in areas hunting.
You know what I decided my first trip out, and not knowing a thing about hunting mountain wilderness in Colorado for Elk? Go Guided. Its expensive sure, but you only have to worry about getting there with your gear. A good outfitter will get you to the Elk, provide your sleeping quarters, food, and any game you get cut up, and packed out. You will likely find it more enjoyable your first time, and alot less work and hassell. Thats just my opinion. Then if you decide to go back another yr.(and you likely will), you'll have alot better knowledge, and idea on what your facing. I consider myself a very avid bowhunter, and turkey hunter in NY. My Colorado hunts for Elk is by far the toughest hunting I've ever done in my life. Its very challenging. Feel free to P.M. if like w/any other questions.