A couple of thoughts:
We were just in the park about a week ago and were surprised by the change in tourists. A large number +50%? or more seem to be Chinese. At least that was our non-scientific impression. This was particularly noticeable in West Yellowstone. For a "cultural experience" go to the supermarket in West and watch the tourists, quite and experience. WATCH OUT for Chinese drivers... wow, apparently there are no rules in Beijing!!!
Also, the Buffalo were caving in the Madison valley, about 10 miles east of the West entrance, along the river. The cows were crossing the river at various places, with their calves - a real spectacle.
A couple of other tips. If you are looking for a free tank dump spot in Belgrade, (just outside Bozeman), go to Rocky Mountain Supply (aka Cenex) gas station and do a fill up. The dump is free.
Also, if you want to overnight, you might enjoy going up Hyalite Canyon south of Bozeman and staying at a camping spot up there. There are reserved spots or you can boondock along the road. The view is pretty spectacular at the reservoir.
Also, if you head to West Yellowstone from Bozeman, via Rt. 191, and want a place to stay overnight, you might try the many trail head parking areas just off the highway (some in the northern corridor of the Park. I can't recall, but they might say No Overnight Parking, but I wouldn't get too worried about that. One that is way off the road is on the west side of the highway, just north of the Park boundary, called Bacon Rind. That would be a good boondock place. Lots of horse outfitters travel out of those trailheads and leave their rigs there for days.
(PM if you have questions / need more info.)
Bob