I've spent more than 20 summers teaching and guiding in the Greater Yellowstone area. Cody is a good base, but it is a bit far from the park. Many choose Gardiner, MT or West Yellowstone, MT. You do have some great things to check out in Cody, though. The Buffalo Bill Historical Center is well worth a visit. Buffalo Bill Historical Center
You might keep it in reserve for a rainy day. Cody also has a nightly rodeo. Cody Nite Rodeo in Cody, Wyoming
In Yellowstone, take a hike out of sight of the parking lot, you'll find wilderness and quiet. Sadly (luckily?) the vast majority of folks never go out of sight of their vehicle. At Old Faithful hike up to the Old Faithful Overlook. You'll see the eruption from a seldom visited vantage point. Do the self-guided tour of Fort Yellowstone at Mammoth, the Ranger Museum at Norris Campground, Norris Geyser Basin (hottest, most geologically active area of the park. Visitor centers at Old Faithful, Canyon and Mammoth have great science and historical exhibits.
If you plan one hike, the best is to the top of Mt. Washburn. From Dunraven Pass, it is a 3 mile hike up an old Model T road. 1,400 elevation gain to a mountain peak. (Everyone should climb a mountain once in their life!) Lots to see along the way, almost always see big horn sheep. At the top is a fire lookout and rest rooms, inside shelter. Take jackets and water, not unusual to have dramatic weather change on top.
Swimming is allowed in Firehole River, but it's not advertised or marked. From Madison Junction go south on one way Firehole Canyon Drive. At almost the end of the drive you will find a large parking area, pit toilets, and a stairway down to the river. The water is warmer than other rivers because of hot springs and geyser run off.
Go to Grand Tetons and the town of Jackson. A hike to Hidden Falls on Jenny Lake is an easy trail. Jackson has lots of rafting outfitters for the Snake River. Fun but not real stressful, my daughter spent her 12th birthday on a raft trip with us, but I've taken hundreds of adults on the same trip that felt it challenging.