We just went a couple weeks ago. We stayed for a week, but you really need a week or two to see everything you want to see.
The main roads that take you throughout Yellowstone are basically a big figure 8. So you should get a map and map out all the places you want to see so you can plan where to go on each day as getting from one side of the park to the other in one day wastes a bunch of time!!! There an interactive map area on Yellowstone's Website that tells you specifically the distance between each of the "main attractions" so to speak.
The mornings and dusk are the best wildlife viewing times and mainly the Lamar and Hayden Valley's to the far north and eastern areas.
Personally, I didn't think too much of Old Faithful, but its worth it to see it once. The kids liked it! I was told later in the week Norris Geyser is much bigger and better, but we never got back to that side to check it out!
Not too far north of Old Faithful is an area called Firepot Falls. It doesn't tell you on the signs, but there is a swimming area there that is pretty cool (you swim right in the river) and the rangers patrol it regularly.
The Grand Canyon of the Yellowstone area is amazing. There are 2 waterfalls there that you can walk pretty close to and one path take you to the top of the falls so you can look over the edge. Pretty breathtaking! Tower falls (further north) was great too.
The petrified tree really wasn't worth the drive. I thought it was going to be a bunch of them. It was just one tree surrounded by a fence!
Mammoth Hot Springs was also pretty crazy. A must see. And that little town of Mammoth was neat too. An Elk and her calf walked right past us!
Bring binoculars if you have some as a lot of the harder to find wildlife isn't right near the road. Also, there's a lot of walking when you go to the pull-offs, so wear good shoes and hats and sunscreen.
We also saw a Grizzy and her cubs up near Lamar Valley. From what I read that's a pretty common area to spot them. (From the safety of your car of course!!!)
They try to sell you a ton of literature to find your way around, but all we really needed was one good map and the book "Watching Yellowstone & Grand Teton Wildlife". Everything else was really a waste of money as they don't tell you about the good secret places (like Firepot falls swimming area)!!
Oh, if you have time, drive down to Grand Teton National Park and check out Jackson Lake and the Tetons. Breathtaking views!!!
Be safe and have fun!!!!