What are "$\mu$-recursive functions" and $\mu$-calculus?
If it is the μ-recursive functions you are interested in (and not the μ-calculus, which is a totally different thing as slimton pointed out) it makes the most sense if you start looking at the primitive recursive functions (PRFs). The set of PRFs is interesting in it self and it turns out that the μ-recursive functions can be introduced by an extension of it.

Here are some slides that seems to be a relatively gentle introduction:
Link