The geyser basin is going off right now! More crazy big geysers acting moderately predictably than in years. My husband goes every year for 3+ weeks to watch geysers. Yesterday he saw the worlds two biggest geysers erupt back to back, Steamboat and Giant. I just got back from a week there, and saw Giant (and it is!), Morning – another of the rare biggies, and spent a crazy afternoon seeing 10 wolves and a grizzly sit in front of my spotting scope for two hours, interacting and eating a bison carcass.
Where are you staying, how long, and what are your interests? It kind of breaks down to Geysers, Wildlife watching, Views and Hiking. The park is crazy big and you can spend far too much time in a car if you don’t realize how far apart things are, and prioritize. If possible and you are staying multiple nights, and want to see it all, I recommend splitting your lodging between the Old Faithful geyser basin and Canyon Lodge at this time of year, if available. Or West Yellowstone and Cook City if you are staying out of the park
Real hikes can be tough as you want to carry bear spray and ideally be with groups that are 4 or more. You can rent bear spray at Canyon. There are several short hikes I can recommend that are well travelled, but I have been far too close to a mama griz with two cubs with no where to retreat (a boardwalk in a swamp in Glacier) to hike without bear spray.
If you want to see wildlife (wolves and grizzly – you can sea bison and elk most places right now) the Hayden valley and the Lamar valley are prime. Also the drive between them is vey scenic – The Grand Canyon of the Yellowstone is a must see - the falls and vistas are amazing - short well travelled but steep walks. And the Dunraven pass is gorgeous. Unfortunately the hike up to Mt Washburn is closed for massive reconstruction. Basically, if you are driving and you see a bunch of people with large telescope-looking things (spotting scopes) all pointed in the same direction and everyone looking intently, find a a safe parking spot and ask them what they are looking at. Don’t waste your time if everyone is looking in opposite directions.
If geysers are your thing https://geysertimes.org/predictions.php Check this when you get to the upper geyser basin. Look and listen for people with radios. They are the gazers. Most are nice a few are cranky because they get tired of answering “when will this go off?” There is no when – just ask if this one has a prediction or when the prediction window is. Bring snacks. The smaller predicatable one have windows +/- 25-30 minutes, the bigger predicable ones can be +/- several hours. Riverside and Daisy are lovely with more frequent and shorter windows, worthat investing a short time – just bring lunch or dinner and enjoy waiting.
I will PM with my number if you have specific questions about any of this. Have a great trip!