Every year here in Utah we read the headlines about out of staters who made misjudgements and spent more time in the mountains than they planned on. Sometimes it turns deadly, sometimes just scary. If you don't know how to take care of yourself in the mountains, book a trip with an outfitter the first couple of times you come out west and learn how to take care of your self.
You may want to book a summer fishing trip for one of your first adventures. They usually cost less than a guided hunt. The weather is warmer and less demanding. Fishing on alpine rivers and lakes is less challenging physically than hunting, and you get to see and experience the mountains of the west. And it's much easier to feel like you were successful. Catch a few fish - your happy. Spend a week hunting and don't see an elk, you feel discouraged.
Decide how you will travel. If driving, you most likely will want to explore opportunities in New Mexico or Colorado since they will save a day or two of driving. If flying, you have more choices of which states to hunt.
Most western states now use a Point system to draw coveted tags, especially for public land. Utah is producing more trophey elk now than most states. But drawing one of the 1200 tags to hunt a mature bull in Utah requires 13-14 preference points and is climbing. If you don't draw a Limited entry tag, you are pretty much limited to spike hunting on public lands. Guides often team with private land owners and sell guide services along with their own allotment of tags. Meaning if you will pay the $3000-$10,000 for their services you can get one of their tags and plan to hunt this year rather than wait until you are lucky enough to draw a coveted tag.
Some states do offer over the counter mature bull tags. But over the counter tags usually equal smaller bulls and more hunting pressure and lower success rates.
Another point to consider. If you do a DIY hunt. How will you handle an elk if you are successful? They are big! I own horses. It cost me a lot of money to keep horses year round. When I go hunting, I want them available for my use. It always bugs me when I'm hunting to have somebody get lucky on a elk came and beg me to help them packtheir elk out. I like to help people. But if I've only got two-three days to hunt myself (then get back to work). I'm not very willing to spend my time and wear out my horses packing out somebody elses kill that they didn't plan for.
I remember a couple of years ago, talking one of my neighbors into applying for an elk tag, he was successfull indrawing a tagand came hunting with us. We helped him to shoot a nice elk. As he approached the downed animal, I watched as he was overwhelmed by where to start. I jumped in and helped him butcher the elk and pack it on my horses. He commented. "I've been hunting elk for years, hiking way back into back country areas with no idea of what was involved IF I ever was successful. Until now I never realized that I would never have been able to retrieve an elk by myself"
If you don't have the resources to pack out your kill, then maybe hiring a guide up front is a prudent thing to do. They know the mountains and can help you stay safe, they have the gear to make you comfortable in the back country, they often have access to tags with out winning the lottery, and they have the resources to get your game out of the mountains.
Like the posters above, I would recommend antelope as a good starting hunt in the west. Mule deer next and Elk as you want more chanllenges. Sheep will be your ultimate challenge. I am fortunate enough to have drawn a Desert Bighorn Sheep tag this year. I am doing it myself with out a guide. It is and will continue to be an extremely tough hunt. Requiring many days of pre-season scouting to learn the area and a lot of days actually hunting. I'm fortunate that for me the hunting unit is just a couple hours drive from home. But somebody from out of state would need a guide. I doubt they would have the time to fly in and spend the time needed for a DIY sheep hunt.
Some pictures from this falls sheep hunting. This area was 100* on Labor day. It was 26* this past Saturday