I'm responding to this because in our COVID world, I'm dreaming of traveling again. My wife and I spent two weeks in Idaho last September. It's now one of my favorite states and I look forward to going back because now that I have some actual knowledge of the place. I echo the comments above. You could fish your entire life and not see everything. The three standouts for me were:
Henry's Fork. We went up to Island Park basically on a whim, and it was cold and pouring down rain. There was virtually no one around. Gorgeous place to fish. I got a recommendation to try the Buffalo River right near there but we got turned around and couldn't find it.
South Fork of the Snake River. I got a guide and a full day float. I'm definitely fishing this river again one day. Definitely recommend checking it out.
Any of the alpine lakes. My wife is not an angler but loves hiking, so I packed a rod on our hikes. I believe many of these fish are stocked but some are wild. Either way, it's an interesting challenge catching fish in water that is so clear it almost looks invisible.
Definitely want to go back, especially to target some of the smaller streams with less pressure. The one big recommendation I'd have is thinking about late August/early September, which is when there are fewer folks out. The weather for us was still in the 60s to low 80s. It's tricky--a week or two after we came home, they got snow.