I agree with Glander, you can see a lot of really nice geysers, mud pots and so forth at the Old Faithful area. If you cross the main highway at Old Faithful, there are spectacularly huge steaming pools. Mammoth Hot Springs can be a definite bust, they often dry up due to seismic activity, but at other times they re-activate and are breathtaking, Mammoth is also out of the way from most areas of the park. Old Faithful area often has buffalo laying right next to the trails. Hayden Valley is wonderful if you are looking for animals. We saw a grizzly bear, a black bear, antelope, a lone wolf trying to get an antelope, elk, bison and eagles all in one day there. Hayden is called the "American Eden".
Note: If you want to see moose, keep looking in the pond and marshy areas, especially if there are young willows growing, they love young willows!
Have a great time, whatever you decide to do!