I am having a really hard time understanding exactly how to determine the Riemann surface of a complex function f(z). I understand the concept: images of these complex functions are periodic, and so they map to the complex plane (not including the branch cut) infinitely many times. Each of these planes is a branch (sub-question here: does it not always map to the whole plane?), and drawing how the f(z) moves from one branch to another is how you show the Riemann surface for f(z).
So given any complex function, how exactly do I find the branch cuts? Is it by checking the zeros or the undefined values of the function? If there are two branch cuts, how do I know which points are part of which branch cuts? (phase factors play a part here, I believe).
Basically, if anyone can give a vague step-by-step of finding the branch points/cuts of an arbitrary function and explain it to a dummy, it would help a lot!