I'll be making my 6th visit to YNP later this year. You can encounter bears just about anywhere in the park, but the Lake and Canyon areas are quite good for grizzlies. Just be careful, and keep a respectable distance -- both for your and the bear's safety! For general wildlife viewing, the Lamar and Hayden Valleys offer a lot of variety -- wolves, bear, bison, elk, prong horns, etc. For moose, you will have much better odds of seeing those animals in the Tetons. There are some moose in Yellowstone, but not nearly as many as before the forest fire in '88. Sightings are not that common.
If you like to hike, I would recommend the Mount Washburn trail -- we usually take the Chittendon Road trail up the mountain, as we've had better luck seeing big horn sheep there than on the Dunraven Trail. The view from the top of the mountain is incredible -- you can see all of Hayden Valley, down into the Canyon, and on a clear day sometimes even the Tetons. Other favorite trails are Trout Lake and Fairy Falls. You get a great view of Grand Prismatic Spring from the Fairy Falls trail.
The geyser basins are popular sights, of course. Check the visitor's center to get an idea of which geysers may go off while you are in that area -- there are more geysers than just Old Faithful.
Grand Geyser is very impressive, but only goes off about once per day. Castle and Daisy are other geysers worth waiting to see erupt.
The Canyon area is my favorite part of the park. The Grand Canyon of the Yellowstone is beautiful. The Lower Falls in the early morning is a great photo opportunity.