Which entrance are you staying in/near?
If you're around West Yellowstone, I think there's quite a few options. The Divide Road/trail is probably the most popular. I've never ridden the rendezvous trails, but they're cross country tracks, so they are probably pretty mellow if the trails are in decent shape. Most of my experience is more around Big Sky area. There are some mountain bike trails in the park as well, but they are mostly just older abandoned roads.
Not sure where you're coming from, but it is bear country. It's unlikely to see bison outside the park, but they can be dangerous as well. It's not uncommon to see both animals using the trails. Carry some bear spray.