You should work in a lab or machine shop, for pretty much the same reasons you give for learning to program. Those skills tend to be under-emphasized in the undergraduate curriculum compared to how useful they are in the real world, so you'll have to seek out opportunities on your own. But if you do, that means you'll be rare and valuable.
To further elaborate the question: What are the most universally and most useful skills one could learn at a university? Currently, I am studying physics as an undergraduate and I am thinking about my career options. I have gotten opinions in other forums as well but I would really like to hear your thoughts and I also think that other LessWrongers would find such a list useful. To give you a rough idea of what I mean I have two examples:
What are some other skills along this line that are universally useful?