I spend the summer on the Yellowstone, close to Emigrant, which iis about the middle of Paradise Valley, south of Livingston. I have fished all of the places mentioned, and all are excellent.
You didn't say whether you are planning to use guides, or have access to boats, which would make a difference of where you might concentrate your time, in my opinion. The Yellowstone, being the biggest, does have access, which of course you will have if you are staying right on the river, but it is best fished from a boat. The same thing with the Madison, but there is more access there.
One river that was left out was the Shields, which enters the Yellowstone several miles east of Livingston. You've got to go way up above Wilsall to get real quality fishing this time of the year, but it has some of the nicest pure strain of Yellowstone Cutthroat of any river in the region, some of which are quite large.
The Boulder is one of my personal favorites in the entire area. I like to fish it above where the West Fork enters it, and if you hike a short distance from a bridge or access point, you can often have the entire place to yourself, and as already mentioned, lots of trout - I caught 3 brown trout there on dry flies earlier this summer in about 10 minutes without hardly moving. The smallest one measured 16".
Two other things you should be aware of right now: smoke and low water levels. Today was one of the smokiest days I've ever seen here from the numerous fires burning in the west. There are not any major fires close to us right now, and who knows, the wind direction might change and clear things out by the time you get here, or it could get worse - just be prepared to live with it when you get here. Also, the water temperatures are okay right now, and should be okay next week, but the rivers are very low for for this time of the year, and the fish have seen heavy fishing pressure for a couple months now and won't be easy to catch.
In any event, I'm sure you will enjoy your stay.
John
