Off and on over the years, I've spent a little time in Cody, only once that I can recall in Powell. I've flown in and driven into Cody and stayed there a few days at a time.
I really second Mike's recommendation to stop at the Museum--it's a multi-day experience for many, because it's so complete. It has something for anyone who has any interest at all in Western history.
The town and the area are really quite pleasant as well. I can't play Chamber of Commerce very well, but as visitors, we've had some good experiences in Cody--good food, things to do, places to see. It's the gateway to the east entrance to Yellowstone, although that's not exactly nearby--about 55 miles if I recall, so a couple of hours, because it's not the kind of highway that you speed along to get from A to B--too much to see along the way, one of the most beautiful drives anywhere in the US.
East of Cody is the Wyoming Veterans Memorial Park, dedicated to Wyoming military who died in the wars from WWI through the current Operation Enduring Freedom. Wife 2 and I visited it on a motorcycle trip that took us to Cody, through Yellowstone, and up north as far as Kalispell, MT. The part of it that is devoted to Viet Nam is a small replica of the Wall in Washington DC. When I saw so many names of those I knew in high school in college, I had a really hard time riding back to Cody--very moving.
Wyoming being the "small city with the suburbs spread out a bit", I have a number of friends and acquaintances in Cody. It's a nice place--one of many in Wyoming that I would consider living in, if I were to leave Colorado.
Cary