We used Cody as a home base to do Yellowstone three years ago. It's about 50 miles to the gate and another 50 to Old Faithful. We did three trips into Yellowstone while we were there. Yes, it was a lot of driving but there was so much to see along the way. We love driving.
Also, we loved the Buffalo Bill Museum. The Night Rodeo was fun. There was also a nightly free wild-west-type show in front of the Irma Hotel.
We also took a trip to Thermopolis to swim in the hot spring pool and sightsee. It's about 1 1/2 hour drive each way.
Another spectacular drive is to take 120 to 308, take 308 over to Red Lodge and then south on 212 (Beartooth Hwy, I think it's called) to the north entrance of the park.
There are a lot of other things to do. You won't get bored.