Despite doing three years of mathematics at university, as well as most of the course work for an honours year and a few graduate courses, I never did a single statistics course. But Camilla's field of research, genomic bioinformatics, involves statistics and she's now working with real statisticians. And recent macroeconomic events have brought me an interest in finance. So...
The next best option to stealing Cosma's brain is to make good use of the online version, and I've been reading through some of the material linked to there. One paper Cosma links to which I particularly like is Judea Pearl's "Causal Inference in Statistics: An Overview" (PDF), which has helped me clarify my unhappiness with so much of the use of statistical methods - in particular regressions - in econometrics and the social sciences. So the new edition of Pearl's Causality is now in my Amazon wishlist, along with Counterfactuals and Causal Inference: Methods and Principles for Social Research which looks like it covers similar material.
Statistics also has connections to one of my first loves, epistemology. On the philosophical side of things, Deborah Mayo's Error and the Growth of Experimental Knowledge sounds fascinating. And I still haven't got my head around the whole frequentist-Bayesian debate. (Update: Stephen Ma recommends Jayne's Probability: The Logic of Science, which would give a Bayesian perspective to balance Mayo.)
On the computer science side of things, Cosma recommends Prediction, Learning, and Games, and there's a free PDF of The Elements of Statistical Learning. (I've studied enough programming languages in my time, but unless they're theoretically interesting I can't see the point of doing that in isolation. So I'll pass on learning R unless or until I have an actual project that needs it.)
Getting back to my mathematical roots, I could try to read Kallenberg's Foundations of Modern Probability in conjunction with Almost None of the Theory of Stochastic Processes. (I have done a graduate course on measure theory and Lebesgue integration, even if it was twenty years ago, so this is not necessarily impossibly hard.)
Apart from Financial Calculus: An Introduction to Derivative Pricing, which is a rigorous derivation of Black-Scholes, I've only read popular books on finance, by Mandelbrot and Taleb and Rebonato (also the last's Coherent Stress Testing). I'm not sure that the mathematics here is that interesting - if anything, I share the concern that abstruse mathematics can hide broader problems. (I remember that one of my tutors warned me never to mix Black Magic and Lebesgue integration, which may explain my uneasiness about Black-Scholes.)
[Note: I managed to post this, without planning it, on World Statistics Day!]