If you're going early-season (May thru the middle of July), your best bet would probably be staying in either West Yellowstone or Gardiner. From West Yellowstone you can hit the Madison, Firehole, Gibbon and Gallatin rivers without a problem. From Gardiner you have easy access to the Madison, Gibbon, Firehole and Gardiner Rivers. In West Yellowstone I'd check out Blue Ribbon Flies (www.blue-ribbon-flies.com) and Arrick's Flyshop, both are excellent and offer guided trips. In Gardiner, I'd check out Park's Fly Shop (www.parksflyshop.com). Runoff during that time would be your biggest issue, the water can be brown from the end of May well into July and is pretty unpredictable; you may want to call into a few flyshops and see what the conditions are before you leave. Yellowstone offers a number of stillwater fishing opportunities if the rivers are blown out.
If you're going later in the season, I'd recommend focusing on the northeast corner of the Park, like Gavin said. Rivers like the Firehole, Gibbon, and Madison tend to shut down as temperatures climb. In the northeast you'll find Slough Creek, Soda Butte Creek, and the Lamar River are the most popular destinations, with lots of native cutts and wild rainbows. They also get pretty crowded during the summer, so be aware of that. There are a few mom-and-pop motels, restaurants, and gas stations in the northeast corner out of Cooke City, but it's pretty sparse.
I'd really recommend picking up the book The Yellowstone Fly Fishing Guide, it gives a pretty detailed rundown of all the Park's fish-holding waters, as well as major hatches and appropriate imitations.
Good luck with your trip!