I live about 9 miles (in a direct line, not by road) off the south-west corner of the Park so I'm quite familiar with it. The main advice you can receive is to use common sense, keep your distance, and don't approach any animals, as already mentioned earlier in this thread. Don't do anything even remotely like the best example I know of how humans are sometimes idiots when it comes to wildlife, the subject of the story being a guy who was parked with a group of cars watching some grizzlys, got out of his van, walked down to the bears and actually touched a cub!!! For some unknown and fantastically lucky reason the mother bear did not attack him but he is incredibly lucky to be alive.
I've only ever been in one worrisome situation on a Park road, that being because whenever there is a wildlife sighting the traffic all comes to a stop and there is a lineup of vehicles on the highway. My wife and I were in such a situation when on our Spyder, the cause being a group of buffalo alongside and on the road. They were some distance ahead of us but one huge bull came walking up the road in our direction, passed about 10 feet from the Spyder with us sitting on it, but didn't even look in our direction and simply walked on up the road, not bothering anyone. But, that did elevate our heart rates a bit! There are usually Park rangers present quite quickly when such traffic jams occur.
Thousands of motorcycle go through the park every summer and I have never heard of an injury or fatality to a motorcyclist, although such may have happened at some time in the past. In some opinions moose are even more dangerous to humans than bears or buffalo, and are notorious for being very unpredictable, and the cows are fierce protectors of their young.
Incidentally, if you are going to or leaving the park by the West Yellowstone gate and traveling south on Highway 20 I would recommend taking state highway 47, the Mesa Falls Scenic Byway, which cuts off Highway 20 in the town of Ashton at it's southern end and cuts off in the Island Park Caldera at it's northern end. It's more scenic than 20, with both Upper and Lower Mesa Falls being very beautiful and well worth stopping to see. This road passes about 4 miles from my house.
Have a great trip and enjoy the many beautiful places you have planned.
Mike
Idaho
www.rtwrider.net