Tyler Cowen invites me to add to a list of experiences that he recommends for a young economics major. His list is a good one.
One thing that I would add is that you spend several months living in a "red-state" community if you come from a "blue-state" community. (If you come from a red-state community, then just being on a college campus gives you a blue-state community experience.)
I think that working on Capitol Hill is a great way to learn the constraints on rationality and decision-making there. When people ask me how I became a libertarian, I cite my brief Senate internship in 1973 as one episode that nudged me in that direction.
Also valuable for me was working at Freddie Mac, and seeing how a large organization, in spite of being profit-driven, is filled with internal conflicts, rivalries, and biases against innovation. You can only live in fear of large corporations if you have never worked in middle management for one of them.
I think that trying to take a new idea to market, both as an entrepreneur and in the context of a large organization, taught me a great deal. Those experiences did much to shape my views about the messiness of the innovation process.
An experience that I never had which I believe would have been valuable would be serving in the military. The reality is that those of us who are highly educated exert considerable influence over the missions assigned to the military, and yet we are disconnected from the day-to-day reality of military service. That is disturbing.