All good advice on the town of Cody. I too would recommend going to Trail Town. Jerimiah Johnson (mountain man) is buried there.
Two out of town trips I always recommend are going over Bear Tooth pass and a trip to see the Tetons.
Another out of the way trip that is interesting is a trip to the old ghost town of Kerwin. It's south of Cody and west of Meeteetse. It's a rough road and 4 wheel drive is recommend but I've made it up there with my 2 wheel drive pickup. A word of caution, you have to ford the river a couple of times and the runoff might be a bit high in June.
