+1 on the recommendation for the Cody Museum, one of the best firearms collections in the world. You'll even get to see some of the shooting irons made famous in TV and movies. You can easily spend a full day in Cody. If you have time another interesting town is Thermopolis, with outstanding hot springs and a bath house where you can soak. There's a big dinosaur museum not too far as well, which the kids thoroughly enjoyed.
On the west side of Cody as you head toward Yellowstone you can stop at Old Trail Town and check out some interesting old log cabins and the gravesite of Jeremiah Johnston, aka "Liver Eatin' Johnson", the mountain man made famous in the Robert Redford movie from the 1970s. A bit further there is a big lake on the south side of the road (Buffalo Bill Reservoir) with an impressive dam and a visitor center that is worth seeing if you are interested in hydroelectric power and the sight of huge volumes of water crashing out in a splendid torrent.
http://www.bbdvc.com/ On one trip we stopped there in the early morning hours before the visitor center was open, and there were a number of bighorn sheep on the rocky hillside only 100 yards or so from us. Have seen a couple of mountain goats in that area before that was the only time we saw bighorns.
Also, it isn't for everybody, but if you have kids and the time to drive down to Rexburg there is a tourist-trap type place called Bear World where the kids can get closer to bears, even bottle-feed bear cubs.
http://www.yellowstonebearworld.com/ It is one of the things that my kids remember most fondly of the trips we have taken up there, they never talk much about the geysers or camping but they sure did enjoy the bears at bear world.
For my money, the most impressive sight up there is the Grand Tetons - just magnificent. The whole area around Yellowstone is just paradise on earth....