My family and I take two trips up to Yellowstone every year. We fish mostly the Madison and the Fire Hole. I don't remember ever having a trip where I didn't catch fish. Usually, about the forth of July, and some time in early Augest. Often the rivers are muddy and high during the first trip, but we still manage to do pretty well. The later trip is usually the better of the two. It's been a pretty dry year, so I doubt that it'll be too much of a problem. Catch a hatch, and your arms will fall off before the fish quit. The smaller waters, of corse, have smaller fish, but I seem to do better on them. The madison, in the evening, produces best for me. I don't fish the yellowstone iteself much, but I wouldn't discourage you from giving it a try. Lots of the little tributary streams may have small fish, topping out at 10" or so, but they have large numbers. It never hurts to pick a little stream, and go for a walk.
If you're looking for other good spots to fish, the mountains around Lander, or Dubois Wy, offer some great opportunities. If you're willing to do some hikeing, you can catch some wonderful scenery, and have a great day of fishing, weather permiting. If you're really lucky, you may even catch a golden trout or two, if your willing to spend half a day hiking. Cody has some good areas for fishing, too, but I havn't spent too much time there. The area near Saratoga has some good little lakes, and if you don't mind small fish, you can catch brookies by the hundred in the streams.
The Wind River/Bighorn provides exelent fishign in spots. I've seen browns up to 12# come out of those waters. Rainbows 9#+ and 5 or 6# cuts. Catching a 3# rainbow is common. Its close, so I know the waters pretty well. I lived in Thermopolis for 5 years, but the first summer I lived there, I walked down to the river, in town, and cought some pretty nice rainbow. They averaged about 1#, but I got some good ones just below the terraces, the largest being about 22".
My advice, plan a rout to drive, through the mountainous areas. Stop at the tackle shops in the areas that look good, and talk to the guys running them. They may not tell you the secret spots the locals fish, but the'll give you enough info to have a great time. Most fo them are just old fishermen who make a living with their favorite hobby, after all.