David,
If you're entering the Park from Gardiner, your best bet is Richard Parks [Parks' Fly Shop, I think] in that town. If you're going in through West Yellowstone, Craig Matthews at Blue Ribbon Flies.
Flies depend on where you plan to fish and when you'll be there. The hopper season is about to begin; when it does that's the dry of choice on every river but the Madison. Caddis is effective also. On the Madison you'll probably have to go subsurface. Check with the fly shops.
Have fun!