It's unclear whether you are experienced in backpacking. Are you asking how is Yellowstone different compared to other locations, or are you asking how to prepare for backpacking in general?
In the first case, Yellowstone would be different than some other places due to:
- geothermal activity. you can encounter hot waters and some of these may have chemicals that are toxic/harmful to your skin, so you wouldn't want to get in them, while other pools and sulfur springs are okay to bathe in. So you would either need to know how to recognize the difference, or stay out of all of them.
- the presence of large animals: bears, wolves and moose. some general knowledge about how to deal with these animals is in order. For example, grizzlies cannot climb trees. Moose can be very volatile.
As for the second case, cannot be answered succinctly here, but I recommend a small book called "The Pocket Survival Guide" by J Wayne Fears (the one with the orange cover.) This is a brief discussion of what equipment and knowledge you need in the backcountry. Useful because it is small (can be taken with you), direct and to the point, and accurate, and of course what you need for survival is a subset of what you need to take with you backpacking.
There's also another small, slim book of similar disposition, aimed more towards backpacking, which I don't remember the name of, but I'll look it up when I get home. If I remember correctly, this one goes more into gear, clothing, etc. Ah! As luck would have it, I just remembered: "Backpacking Tips: Trail-Tested Wisdom From Falconguide Authors (Kestrel) by Bill Schneider, Russ Schneider and Laura Zorch". There are a lot of nuggets of wisdom here.