Just to get you started:

My info is from like a decade ago when I dated a girl who lived there, so I could be off. But from what I remember, cross bike would absolutely be my pick. You'd be giving up a lot of great singletrack by going roadie only.

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JustinDoesTriathlon

Owner, FuelRodz Endurance.

It's awesome. Bring your road bike and rent a MTB. Ride from Ketchum to the Galena Pass and back - a great climb and rewarding decent. MTBing is all around you but a cross bike won't cut it - climbs can be steep. Great fly fishing as well!

If I could only bring one bike I would say a gravel bike. One of my favorite rides in the area is the Harriman trail to Galena Summit lodge and back. You could also ride Trail Creek summit was is part of Rebecca Rusch Private Idaho gravel ride

Amazing offroad/gravel riding. Agree a gravel bike would be best as many of the paved roads end in dirt or gravel.

Plenty of trails for running

Road riding is pretty good too, although limited in selection. There is a single road running upvalley that takes you from Ketchum north to Galena Pass (great climb), then on to Stanley and beyond. Its reasonably safe for riding early in the morning with a shoulder about half way to Galena. Going south you can take the bike path or road to the town of Hailey and beyond. There are a few side roads. But selection is a little limited compared to other places you can visit.