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?
As @EpiGrad, @JeffE, and @Swiss Coder said, you should learn about subjects that are outside the direct focus of your research. This provides you with knowledge and tools you might otherwise never obtain. I would add though, that it is rarely wise to take a course in something you have no interest in simply for the sake of "knowledge". However, if you are truly interested in something, don't scrap the possibility of taking the course just because it is not directly part of your research topic. Higher education is intended to help us become well-rounded humans,not force us into the narrow trench of knowing only about a specific field.
Also consider, of course, whether you really need to take a course on this, or if there are other ways to learn what you want to know. Auditing a course is a good option, especially if you don't need to receive credit for it. There are also numerous online options for learning,on your own time and without tuition costs. For myself, though, online learning without the support of a professor and peers rarely works well. I need the motivation of knowing that someone is tracking my progress.
So definitely explore subjects outside "your" area of expertise, and keep in mind that a formal course may not be the only or best option.