I live 60 miles from Cody. Here's my list. Many were mentioned already.
Good meals at Proud Cut, the wife and my favorite place to eat. Cassie's is good but a bit pricey.
Bubba's BBQ in Jackson is much better than Cody's.

Must see....
The Museum.
The Beartooth and Chief Joseph highways.
North and South Fork highways.
The parks, of course, Yellowstone and Grand Teton, but a bit too many people for me in peak season, which will include your trip.
Sleeper alert: Bighorn Canyon National Park, 60 miles east of Cody. No crowds, great views. Take the boat trip at Horseshoe Bend. Or PM me for an even better boat ride, if I'm not fishing somewhere.