I am a graduate student in physics. One of my biggest frustrations in my education is that I often find that my mathematical background is lacking for the work I do. Sure I can make calculations adequately, well enough to even do well in my courses, but I don't feel like I really understand what's going on. To combat this problem I have decided to learn mathematics a bit more rigorously. At this point I would like to learn a bit of differential geometry, some abstract algebra, and some functional analysis. The problem is, I don't really know where to begin. As an undergraduate I took linear algebra, ODEs, PDEs, and vector analysis. Those were more or less the only courses I took beyond Calculus I-III. As a graduate student I have taken your typical math methods course out of Arfken and Weber. These courses have not prepared me to read a book written for mathematicians. I recently picked up Bishop and Goldberg's Tensor Analysis on Manifolds, however the book looks quite daunting to me. It is in a language I am not entirely familiar with. My question is what are the mathematical prerequisites to begin reading a book such as that. What should I read before I ever pick up these math books? Is there a quick intro into the language of mathematicians for physicists?