For those into physics, I wholeheartedly recommend Marsden and Ratiu's book, "Introduction to Mechanics and Symmetry", which deals mainly with the different formulations of physics applied to symplectic and associated geometries.
Even as a hydrodynamics software guy, I found the computer graphics research community to be the easiest entry-point for, especially, the topology and modern differential geometry. It's especially nice when they do a simulation paper with a high end geometric/analytic approach.
This might be a good place to go in order to have a start at, say, Arnold's ``topological methods in hydrodynamics'' or anything TQFT-esque.