Just start somewhere (say Kaggle/Scikits) and then just work your way through. Many will tell you that you need to know 2^{X} to get somewhere, but what they mean is that some elementary familiarity with some of X is advisable. The exceptions are calculus, and some vector/matrix math (really the definitions in "linear algebra").

The math for things that work (these days) ^(TM) is quite boring and would take a quarter of lectures (see any of a number of MOOCs). Some of it is quite tricky like convex optimization and graphical models, but you'll likely never have to know these details unless you're in academia. The latter have also fallen out of popularity, so you know... less things to know. Convex optimization these days is all about stochastic-distributed things, which is probably not the ML you (or most "data scientists") have in mind.

Best thing to do would be to work through the homeworks of some course offering at CMU/Stanford/UW. The principle is generally simple, but you'll have to wade through the symbols, and implement something to make sure you understand the symbols. This is probably the best exercise if you want to understand things.

You could also just look at and modify the code in scikits, since this will save you a lot of work with regards to the engineering bit. Kaggle might be help, but I found them mind-numbingly boring (usually with leaky datasets), which have to be hacked for performance with endless tweaking on XGboost.

To be honest, I really think ML is overhyped (it is also quite boring as a job). If you want to do math, you should be doing math.