The best way to become familiar with BUGS is to look through examples of simple data modelling problems similar to the ones you are interested in. Browse the examples pages to see the huge variety of models that can be implemented. Look at one or two in depth.
To understand the background of how BUGS works, please read the chapter on Monte Carlo methods in Information Theory, Inference, and Learning Algorithms. BUGS uses a method called Gibbs sampling.
There are two versions of BUGS, both of them free: WinBUGS was developed at MRC-BSU in Cambridge; OpenBUGS is a GPL version that works under linux. I have installed OpenBUGS in my ~/BUGS directory at AIMS (following Iain Murray's instructions). The documentation for WinBUGS and OpenBUGS are both highly recommended when you are using OpenBUGS.