Having been to Yellowstone most recently in winter (New Years 2008) we decided we'd go in Spring and see what wildlife we could see before high season began. The timing turned out well - we planned to leave as Memorial Day Weekend began and had several days of viewing before then. As we were leaving the line of cars to get into the park was so long we were really, really happy to be getting out then! So if you plan a trip, it might be a good idea to aim for the shoulder season of early spring or late fall, because the frustrations of dealing with cars and people really impact the pleasure of wildlife viewing.
If animals are your focus, plan to spend the most time in the Lamar Valley. The closest place to stay and be in the valley for early viewing at first light is Cook City, MT, which has a few (very few) places you can get a room. The second best option is Gardiner, MT near the north entrance, and the third choice would be West Yellowstone. This trip we were in Gardiner. We intended to spend most of our time in the Lamar Valley with a few trips to Hayden Valley. We ended up doing all the open roads in the park as their were road closures in two sections, making us take the long way around. By far the most wildlife was in the Lamar Valley with only a bit in Hayden and little to none elsewhere.
If you plan some hikes, beware that many trails are closed due to kills near the trails that attract bears, so check the ranger station for open trails, closures, and data on road closures - due to construction, flooding, or snow yet to be removed. The Tower to Canyon road is not open in early spring, fyi.
If at all possible, bring a camera, binoculars for all people traveling, a cooler of beverages so you don't have to stop/shop in the park, sunscreen (you'll burn fast at these elevations), fleece and gloves, etc. for warmth in the morning and evening when it is still frosty, and a spotting scope with tripod. We did not have a spotting scope but knew it was needed from a past trip, so we posted an ad on Craigslist and rented one for 10 days from a local guy for $25 - well worth it!!! There are things you simply cannot see without a scope, and the same is true for binoculars.
To see the best stuff, you have to get up early to be in the Lamar Valley at first light. That's your best chance to catch wolves, and to see large carnivores on a kill. That means getting up at 4am if you are in Gardiner, on the road by 4:30am - it takes an hour to get to the valley and first light is around 5:30am. If you stay in Cook City you can probably safely leave at 5:00am. In the area closest to Cook City, look for moose. They like the willows in that area, between the east end of the Lamar Valley and the east edge of the park. Coyotes are easily seen in the morning. Prarie dogs are in sagebrush everywhere if you sit still a minute and wait for them to appear. Look on the north side of the car whenever there is water near the car for beaver. Look at the treeline in the valley for black bears, and out in the open for grizzlies. Bears are often simply small dots, so without a scope you can't see them at all. Sometimes you can see one close but more often they are far away. Wolves can sometimes be seen with the eye or binoculars but a scope is often necessary there too. The second best time for viewing is in the evening from about 6pm to 9pm. During mid-day many of the animals are resting and most carnivores are inactive.
I'll be posting our favorite photos in the days and weeks to come. Overall it was a great trip - we were lucky enough to see wolves several times, several bears, baby bison (so cute!), lots of elk, deer and bison, tons of birds, as well as beaver, badger, prarie dogs, pika..it was very good.