Equipment needed for watching Yellowstone wildlife should include a spotting scope, which can be rented in nearby towns (supplies limited, so book ahead). If you want to take photos, get a long lens with a wide aperture (preferably at least 2.8 f-stop for low light situations). You can rent or buy an extender from photography suppliers that will increase your range but decrease your aperture some.
A sturdy tripod to set your camera on helps. You can go hours without action and then have a flurry of things happen quickly. Practice your setup and taking apart before you go.
Temperatures can vary extremely, so dress in layers.
Most of the regular watchers are extremely helpful and will point out where various wildlife can be seen. It’s better to ask them questions when they aren’t busy.
If you’re driving, try to park where other people will be able to get in and out — there’s limited room at the pullouts, and when something exciting is sighted, they can fill up fast.