I usually make one or two trips up to Yellowstone each summer. What do you want to know?
We do one ride that we call our Buffalo ride. We park in the Hayden Valley and ride back into Mary Lake, There are always 300-400 buffalo back in there. We occassionally see a bear. Always see bear tracks on trail.
The Hayden Valley is just rolling hills. It is an easy ride, But better suited for late summer, There are numerous boggy areas, that are easier to cross when they have dried out in Aug .
Another popular trail is into Union Falls. Again it is pretty flat, but it travels along through the timber and follows several rivers. Flies can be real bad too early in season. Dust along the trail can be bad if you have a larger group late in the season.
You do have to cross some pretty good sized rivers
The biggest challenge you will have is Where to Camp with your horses. You will basically need to camp out side the park and then trailer into what ever trail head you want to ride for the day. The area near West Yellowstone and Jackson Hole are difficult to find horse camping. But there are places to camp 20-30 miles down the ride. I have not entered the park from the Cody or Gardner entrances. So my experience is limited to the South and West gate.
Yellowstone is really fussy about the possibility of transporting any Non-Native seeds into the park. If you bring feed with you, its probably best to bring hay pellets.