You're in a place with geysers, paint pots, incredibly colored thermal pools, hot springs, lakes, hiking trails, waterfalls, woods, mountains, valleys, and critters of all sizes. Welcome to Yellowstone National Park - the world's first National Park - and what a park it is. It's huge and truly one of a kind.
You've driven (or flown) for miles to get here and seen some of the most rural areas the US has, yet, despite its ruralness, this park is such a gem that it always has a crowd in summer and fall. So many people want to see it - after seeing it yourself, you can't blame them. The sights are simply incomparable.
Note, it's also mostly about 7500 feet in elevation, so expect a chill at night - even in the dead of summer, yet it's very hot during the day. Prepare for 30's to 90's.
So, where to start? To best see Yellowstone, first, do some research. Seeing it all in one or two days isn't going to happen, so a minimum of a couple nights lodging is recommended. We stayed 4 nights and could have used at least one more. Decide if you want to camp or stay in a lodge/cabin - you'll most likely need reservations well in advance. (Some camping is first come, first served, but you still generally need to get there early.)
If you're on time constraints, you'll have to decide what's most important to you to see. Simply driving from one end of the park to the other will take you 2-3 hours (one way) - without stopping - and if you don't stop, did you see anything? Some stopping (for critter jams) may be required. "Critters" can be anything from buffalo/bison, elk, bears, moose, or anything else others consider worth stopping for.
For quick advice, for thermal features, the south western part of the park is best - for critter watching, the mid-eastern tends to be better (though critters can be seen anywhere, so keep your camera ready!) Waterfalls are (mostly) in the east/northeast. Mountains are everywhere as are hikes.
To allow some space to actually write, all of our Yellowstone experiences are in separate journal entries. This is mainly an overview to "warn" you not to underestimate the time for this park - especially if you want to hike. (This is our second time there after underestimating it the first time...) There are more geysers than Old Faithful, so even that's not a "short stop just to see it" for most folks. The pools are so incredibly beautiful that they too take longer than most anticipate. Then there's the animals and "critter watching" - a whole fun aspect in itself. Read on to see what might interest you!
There's food, gas, and lodging in this park, but expect to pay extra for all of the above... though the cost for camp sites was quite reasonable.