satt comments on The "best" mathematically-informed topics? - Less Wrong Discussion
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Something to consider: what's a good field in which to learn basic statistics (sticking with the "learn by doing, when possible" theme) ?
I'll side-step the "field" part of the question and instead point to an undergraduate lecture course on data analysis which has some online notes and a series of exercises.
It's worth pointing to something specific one could immediately start working on, because I think people underrate the trivial inconvenience of not knowing which specific book or course to consult. The course linked is not that basic, admittedly, but even if it's too advanced it should help highlight specific keywords & terms to look up on Google, Wikipedia, or textbooks.