First, I say be careful camping roadside. I'm pretty sure you can't do that inside Ystone itself, and it would scare me ****less to have wolves or bison sniffing around at 3 a.m.
Western rivers are very different from anything here in TN. I say get a guide for one day on each water. You don't have to, but if you don't, you may go home wondering why everyone raves so much about them. The biggest difference to me is that you throw these freakin huge, gaudy flies MOST of the time, in some of the rivers. And the others, like Henry's Fork, have huge, huge trout that are very well educated in many sections. I say forget Henry's Fork unless you have a guide.
If you are wanting to hit the big names, then be sure to hit the Madison, Yellowstone below the park, Yellowstone in the park, Henry's Fork, and Gallatin as it goes thru the park. The Firehole is really neat if the water isn't too hot yet. You'll have to go by fly shops and get status reports on all the rivers. The Lamar can produce large numbers of huge trout if the water is right, but most often it is too high and muddy. You really should get a guide and raft the Madison, Box Canyon of Henry's, and the lower Ystone. Don't forget to hit a few of the smaller waters in the park as well; you've already got the guide books for that. Trout Lake was mentioned above, and all I can say is that I have seen a trout the size of a coyote in there. I think they are very hard to catch. It would be more worth the quick walk in to Trout Lake, walk round it fast while looking for trout, and if nothing seems to be hitting on top, call it a good hike and go elsewhere. If there is a log still near the feeder creek, walk up on it slowly, as that is where I saw the huge one 2 years ago.
There is so much good water within 100 miles of Yellowstone. I have only fished a small part of it. My first trip there 10 years ago was the only time I visited somewhere and imediately considered dumping everything else in the rest of my life just to live there making minimum wage or something. I think it must be the most special place in America.
Some above have mentioned Slough Creek. It is great, and it can be a long hike to get away from people. I say if you see a stretch of it from the road that doesn't have people on it, just go fish that spot. Same thing for Lamar if it is in good shape. Gosh, forgot about Soda Butte, which can be great and probably crowded too when it is. I should say now that a Westerner's concept of "crowded" is much stronger than what it is here in the East. In some places, people will get p****d if you get within 200 yards of them. Others, like downstream of Fishing Bridge (a place you HAVE to see), you may be fishing on top of each other. Just don't let it bother you if you get yelled at.
Oh, and make sure you stay well versed on the regs. Many waters are off limits for one reason or another.
Bear spray and bells are good. I've seen several bears, but no grizzly yet. This guy was on the way back from Slough Creek:
Drink plenty of water, but be sure to carry some junk food. Dehydration is a problem, but hyponetremia (too little salt in the blood, most often caused by drinking too much water and sweating too much) can be a bigger problem now that most people know about dehydration. I disagree a bit with Kytroutbum's advice about drinking so much water, especially in conjunction with taking so much Ibuprofen. Just drink plenty of water, and carry some salty snacks like crackers or candy bars or gatorade or gorp, or my favorite, chedder cheese with crackers and pepperoni. Dehydration is more common, but hyponatremia is much more dangerous. I read that the rangers in the Grand Canyon now consider hyponatremia to be as big a problem as dehydration. Here is little more info on this:
Take your time. Yellowstone roads really get gummed up during the day. If everyone else is blocking the road looking at Elk, you may as well do it too.
Mosquitoes will devastate you in some areas. I counted 20 on my uncle's back, which was the only part on him not covered in deet, in one of Slough's meadows.
Have fun! You'll have no choice but to.