You will learn whatever programming you choose to learn. I don't think there is anything about theoretical physics that naturally requires programming. You will learn how to use Mathematica or Maple (or both), and do a little programming in those, but that is not very transferable. You may find yourself doing numerical calculations, but in order to perform them you will most likely use Mathematica/Maple rather than writing your own numerical algorithms from scratch (which would be a silly thing to do, given the expertise that has gone into existing mathematical software).
If you do some kinds of simulations, then you will do some possibly intense programming, and you will learn all kinds of useful algorithms that have been developed by others. It is unlikely you will come up with a new algorithm that is any better, because these problems have been thought about for a while; but if your simulation is sufficiently unusual, maybe you will find something new. At any rate, programming on simulations is quite transferable to other fields such as aerospace, engineering, and video games. You probably won't write any simulations in a field like string theory, but you will probably do simulations in fields like biophysics and materials science (biophysics is really hot right now).
However, it is very unlikely you will have to do anything remotely related to web design in the course of physics research. Web design requires an entirely different set of programming skills. You will need to learn those skills separately if that's what you're interested in doing.