Hwy 20. This is the East Entrance to Yellowstone beginning in historic Cody, WY terminating at the YS Grand Loop junction. Obviously this road can be ridden west to east but the scenery and fun factor builds better coming in from Cody. I dont recall a lot of services along the route so make sure you top off in Cody.
The scenery is great and the road spends a lot of time winding along the Shoshone River. A few miles after you enter the park the forest will open up to some wetlands on the south side of the road -- keep an eye open for grizzlies rooting around in the grass.