I just got back from there last month. Some must sees are (obviously Old Faithful, but it wasn't my favorite geyser) Yellowstone Inn., Yellowstone lake (hard to miss), and ALL of the falls (the falls are just amazing!), Mammoth Springs (when I was there, there wasn't any water flowing through them so they weren't as cool as the pictures), actually the whole Mammoth area at the North end of the park is cool, the Grand Prismatic Pool, and the West Thumb Geyser.
My favorite geysers were the Beehive Geyser, and another one that started with an "a" (it was called anenomie, or something close to it)
Most likely the three most common animals you will see are Bear, Bison, and Elk (And next month the Elk will have their antlers mostly grown out, and starting to get rid of their velvet) I never saw a Wolf, or Moose. Saw a few Coyotes, but only 3. I only saw 2 Bears but it still had about 8 feet of snow in some parts of the park.
Also, don't be rude like other people are, and stop in the middle of the road to take pictures of animals (unless it's a Wolf or something you might not see a lot.) Definitely DO NOT stop for Bison, unless you're at a pull out or something. Because you'll see hundreds. (I saw AT LEAST 200 Bison, and AT LEAST 100 Elk, but only about 17 Elk were Bull Elk [the kind that have antlers])
I highly recommend going through the town of West Yellowstone. That's a nice little town. I went in an Old Fashioned Candy shop that also made fudge, and... OH MY GOD! I don't like fudge but I liked theirs. (even if it might have been made with a mix) It has a lot of T-Shirt shops too.
And over by the Yellowstone Inn. by the Old Faithful there is a fast food restaurant that I liked. I think it was called The Geyser Grill, it was really good. I recommend the Chili Fries, unless you don't like Chili.... or Fries, lol.
And if you want to get Cell Phone reception drive along the lake and they are a few pullouts that have a few bars (I have AT&T)
Anyway, I hoped this helped, it's really a nice place, but I don't like their road signs, it's easy to miss some roads. They also have cool stores (By the way every store is the same, lol. Just with different layouts maybe.)
Gladly been there!