So many options out west, but I agree with David.
Yellowstone is probably at the top of most flyfishermen's bucket list. Although, his recommendation for Durango is a good one, and in fact is a favorite fishing destination for my wife and I. Depending on time of year you plan to go will also dictate where to go. Durango is typically fishable earlier in the year than places in northern CO, WY, or MT due to the climate. If you won't be there until June/July then you have many more options.