"Oh give me a home, where the buffalo roam, and the deer and the antelope play..." Wyoming is that place (or could be anyway) and Yellowstone National Park is one of the prime spots to see not only those, but also elk, coyotes, bear, wolves, and birds of all shapes and sizes... Besides seeing the thermal features (Old Faithful, etc), seeing the large animals most of us so rarely see at home is one of the biggest highlights of this park. Here are a couple tips we've found by both experience and asking some of the rangers.
Tip #1: Critters can be seen anywhere at anytime, so ALWAYS have your camera ready. Some of the best shots are candid, unexpected, sights.
Tip #2: Critter "hunting" is best done in the early morning or later afternoon/early evening as that's when the animals are most active. We opted to go critter hunting each evening we were there - and were never disappointed.
Tip #3: These critters are WILD and can be DANGEROUS - and if you're dumb and get too close so they hurt you, they get killed for it. Please, give them their space. This isn't a zoo or Disney World. You're in their turf - enjoy them from a respectable distance. Buffalo are bigger, faster, and can and do hurt tourists.
Now, for specific locations, etc to "hunt." (Where you have your best odds to see certain types for your advanced planning purposes.)
For buffalo/bison, check out the Hayden Valley just north of Sulfur Caldron or the Pelican Valley just east of Fishing Bridge - esp in the later afternoon if you'd like the possibility of being in a "buffalo jam" (when the buffalo are on the road requiring traffic to stop). At other times they can be seen in the fields grazing or sleeping. In the Hayden Valley they can often be seen crossing the river (swimming) - which alone - is a treat for many of us. There are also a couple near the thermal features in the western sections of the park, but the larger herds are in the eastern areas.
Elk are all over - they can be seen in the eastern valleys, but are more often up close near the road or on some of the hikes in the western section of the park. At times there will be "elk jams" but those aren't as often from elk on the roads as they are from people stopped admiring the huge racks on the bulls.
Coyotes are all over too - we saw ours between Norris and Old Faithful hunting in the fields - you had to look carefully for motion, so drive slowly (or stop at the pullouts and look).
Yellowstone has wolves too - and they are moving out to more areas so check with the visitor's centers to see where you might see them. They're still pretty reclusive. People were stopped at one of the overlooks in the Hayden Valley looking for them in the evenings when we were there, but one needed some powerful binoculars to see them. Look closer in the water close to you there and you can often see a beaver and sometimes pelicans. Beware when you take pictures of a "doglike" critter. Some folks had pictures of coyotes they were calling wolves... If in doubt, ask a ranger.
Bears... our nemesis here... we went looking for them - they are most often found in the northeast between Canyon and around Tower-Roosevelt - or at Fishing Bridge Campground - but we always came up empty this trip. The rangers told us in August the bears like to go farther in the back-country hunting the ripe berries. He suggested coming in July to have better odds. On our trip 2 years ago we did see one down in Grand Teton National Park - eating berries on a hill next to the road. Still, you could get lucky. If you see cars pulled off looking at something, stop and see what it is. Many times that's how bears are spotted - one family saw it cross the road, the rest saw their car stopped.
We hunted near Yellowstone Lake for moose, but never saw any. A ranger suggested the best tip for seeing moose was to go to Grand Teton National Park. We did see them there two years ago.
Birds of all shapes and sizes can be found anywhere - esp near water. We saw pelicans, geese, storks, cranes, herons, trumpeter swans, woodpeckers, osprey, and various assorted songbirds. Then there's the ravens... beware when camping, they're watching you and they like people food and aren't afraid to raid yours if you take your eyes off it.