Should I take courses that are not directly part of my research topic? I've received conflicting advice as to whether I should choose my coursework based on my research topic or not. The main pro usually is that I'll quickly be able to get up to speed on my research. The con is usually that there are so many other courses that I could take, in which I could learn topics I may not ever have a chance to formally learn, and given the constant need to hunt for funding, I may never have a chance to put aside and study in-depth again. Any definitive answers to this topic?
I don't think a "definitive" answer is possible, but the following is based on personal experience and observation of many other students.

If your advisor is okay with it, take as many courses as you can in things that interest you and are in the realm of your discipline.  As an applied math grad student, one of the best things I did was to take a graduate course in optimization from the CS department, even though I thought it had nothing to do with my thesis (in numerical discretization of PDEs).  It ended up being crucial and allowing me to publish at least one paper that I never would have written if I hadn't taken that course.  I also took courses in things like astrophysics, quantum mechanics, and turbulence; I don't use those things much but I can converse with scientists in those fields, which is often useful.

Of course, I didn't take, say, philosophy or Italian or business management courses -- stick to courses related to your field.  And make sure that whoever is paying you is okay with it.