If you do go to Yellowstone, from Cody instead of heading directly east into the parkl I would suggest going over the Chief Joesph Scenic Highway to Cooke City and the northeast entry. Beautiful drive and Cooke City is a cool little town.
From there you can also head over the Beartooth Highway to Red lodge, MT; this is considered one of the most spectacular drives in the country. Much of it is above timberline and there is a good chance to see mountain goats.