I'll second what's been said about learning a language like C++ or fortran if you want to run large mathematical simulations. Pretty much everything in astronomy is written in one of those two. That being so, the amount of programing skills you actually need are pretty minimal. After all, most astro codes don't have user interfaces or display output; they just do lots of math over and over again. You probably could go far just knowing how to structure loops, write conditionals, and read and write to a file.
Knowing what math you want to computer to do, however, is a separate question.
Anyways, if you know the physics you'll be using, all you really need is a decent desktop running some linux or unix variant, a copy of gcc or gfortran (or some version of this that works on Windows), and some patience. Unless you want to do *really* detailed projects like supernova explosions or giant planet atmosphere models (which would require a supercomputer, or a cluster, at the least), you're looking at maybe 100 hours of CPU time, on the longer end. Most of the stuff I do takes shorter than 2 hours to complete on my desktop.
As an example of a CPU intensive code, check out John Chamber's Mercury integrator (
http://www.arm.ac.uk/~jec/), which handles orbital dynamics, and has been used to model planet accretion.
Edit: Also, the book Numerical Recipes is great for looking up ways of doing standard routines (like integrating, parameter fitting, etc.)