The following is common knowledge. 

In the fall, starting about Sept. 15, the browns will start to run up the Madison in Yellowstone.   They will  run all the way to the and maybe beyond the junction of the Gibbon and Firehole.  So, many people, including myself, go to Yellowstone in late Sept and early Oct to fish for them.  The fly of choice is the woolly bugger black or black/olive, which is a streamer, and the rubber leg and copper john, which are nymths.  

The more you fly fish, the more you learn and the more your skill increases.  Join your local Trout Unlimited or Federation of Fly Fishers.