A few things to remember that might be different in the Wyoming and Montana Rockies (at least compared to Ireland!):
Altitude--plan to take it really easy the first few days until your body adjusts a bit. If you can start in a place where you ascend gradually, or "climb high sleep low" for a few days, so much the better. Know the symptoms of altitude sickness so you can descend if they hit. Everyone is different in how they adjust to high altitude, and fitness seems to have little to do with it. Having grown up in Wyoming, I never paid any attention to altitude. After many years of living at or close to sea level, though, I found the hard way that I do need some acclimatization time. I don't get sick but I can only do about half my normal mileage the first couple of days.
Snow is possible in the Rockies any month of the year--two years ago it snowed in the Winds Aug. 8 and 15. If my dog hadn't gotten sick, I'd have been camped at close to 11,000 feet for the second one. It's a good idea to have a weather contingency day or two in your schedule. If you don't need it, you can always hole up for a day for rest, exploration, fishing, whatever in an extra scenic spot! Nights at high altitude even in midsummer are often frosty. Be prepared!
Almost daily thunderstorms. This basically means you need to get up really early so most of your hiking is done, or at least you're off exposed areas, by about 1-2 pm. The storms may last 30 minutes (often less) and there is usually an hour or two break between, but you don't want to be up on a high plateau or pass when lightning is going on. The climate has changed since I grew up there, when the thunderstorms pretty much went away by early August. Now they last until or into September!
Some possibly dicey stream fords, especially if you come early ("early" being June and early July while the snow is still melting). Glacial streams, usually higher on warm summer afternoons, are best forded in the morning when water is lower.
Glaciers themselves you probably want to stay clear of, unless you have lots of experience judging glacier conditions and have the proper gear and training. This is especially true during the warm months when they are melting and moulins (drain holes) are a hazard. If your trip is mostly on trail, not an issue.
Read up about behavior around bears and how to secure your food. It's pretty simple. Just don't panic about it.
It sounds as though you're getting a great trip lined up! I'm excited for you!