Yellowstone is a wildlife and natural wonder paradise. One one hand you've got geysers and hot springs worthy of a visit, and on the other you have some of the most accessible wildlife in all of North America.
Now, don't be fooled about doing a 1-2 day quick trip here. This place is HUGE. I've been to many parks, and was glad I had 4 full days for my first trip. I just scraped the surface in terms of seeing geysers and hiking trails, but managed to drive the 'loops' for amazing wildlife spotting.
The two best spots for potentially seeing the main draws - wolves and grizzlies, plus other wildlife are Lamar Valley and Hayden Valley.
If you're heading to Lamar Valley, the Northeast Pass is a beautiful drive to Cooke City. Try to stay at Roosevelt if you want ot spend time in that area, it's quick access, with great food at the lodge. (Great is relative, as the food options are mediocre for the most part, as the accom and food is controlled by Xanterra).
It can be crowded in the summer, but because it is so big, not a big deal.
For a first visit, drive the loops and check out Old Faithful or Morning Glory or other geysers. IMO Old Faithful is not the most impressive, but it is famous.
If you ahve longer, do some hiking and visit Lower/Upper Falls or Mt. Washburn.
To enjoy it all, you definitely need to stay a week or longer! If camping, sites can book up fast, but even in summer you can usually snatch one up without a reservation at some less popular campsites such as Norris, but watch out for Bison at night, they roam free and are hard to see!
The drive from Mammoth to Tower Falls is a great one too, and Mammoth Hot Springs is a great little town with Elk everywhere, it is surreal.
Just remember that these animals are totally wild. Even though Bison are everywhere, they are fast and can be aggressive. People take this for granted every year and suffer the consequences.
Head to the Tetons further south if you have time!

The first and one of the best National Parks in the United States! The scenery is unbeatable and the wildlife viewing opportunities unparalleled.  If you get a chance to visit Yellowstone I recommend either doing a back-country trip or going during the winter.  Getting just a little off the beaten path will give you a chance to really experience Yellowstone's wilderness without all the crowds.  And, during the winter the park is nearly empty and the wildlife is easily visible, we saw fox, bison, wolves, etc.  There are opportunities for cross-country skiing and snow-shoeing.  The geysers in the winter are also quite a treat.  There is also the opportunity to visit some of the hot springs, while it is snowing out!! Try the Boiling River.  Absolutely fantastic!

I loved going to Yellowstone. There is just so much to see and take in. Can't believe that all i say was unaltered and straight from mother nature. I love seeing the hot springs and all the geysers. The best part of my trip was to see the water falls...you feel so small in this world when you stand there in awe. I was glad to see that new trees are growing after the fire they had years ago. I suggest taking a map or GPS when going thought Yellow Stone. I went on a bus tour , but we ran into a lot of lost people. Also Pack some snacks and alot of water, you never know how far your hike will be so be prepared. If your not camping in the park, make sure to leave before sundown..it gets dark...really really dark and help getting lost is not fun at all. have fun and wear good shoes..no one to impress,but your feet.

After the Grand Canyon, this may be one of America's best natural attractions. I took my parents there a few years ago. About half of all the world's hot water displays are here. The rest are mostly in Iceland or the Kamchatka area of Russia. There is a natural hot spot here -- a massive volcano. It is also America's first National Park, and an inspiration for the entire world. For example, Volcanoes National Park, mostly in Rwanda today, was established by Belgium's King Albert, in part in admiration of Yellowstone.

Yellowstone is one of the most pristine places in the country. It's 2.5 million acres of untouched beauty and nature. There is so much to do there, you can hike, camp, take guided tours, fish, and there are even a few places to swim. They offer horseback riding tours and bike rides. It's amazing to see an entire eco-system at work, what with all of the wildlife, hot pools and thermal features. It's a great place to take kids to give them a respect for nature and wildlife. I worked here for 6 months in 2004, and it was the most amazing time of my life.