We camped at Bakers Hole campground and entered the park via West Yellowstone. We used the camp as our base and did different directions in the park each day, we did end up back tracking sometimes but it worked for us. The only snake we saw was a teensy little guy right on the trail. We enjoyed all the "normal" tourist things, but you always need to allow for people who stop in the middle of the road to take pictures of coyotes and buffalo. We hiked a few of the trails and did have snow in June 2012. We were lucky to see lots of Elk and a few bull moose, deer and pronghorns. We spent quite a long time in the Grizzly and Wolf Habitat in West Yellowstone area, check the times of the shows and the times for feeding, very interesting.
We have been fortunate to visit other areas with wildlife but if you've never had the chance, I think it is very worthwhile.
We headed south to the Grand Tetons after our few days touring Yellowstone. I absolutely loved the mountains...and the storms.
Enjoy your trip!