Biology is really about stats. Indeed, the foundation of much of frequentist statistics - RA Fisher and colleagues - were totally motivated by biological problems. We just lost the link the heyday of molecular biology when you could get away with n=2 (or n=1!) experiments due the "large effect size" - ie, band is either there or not - style of experiment. But now we're back to working out far more intertwined and subtle things. So - every biologists - molecular or not - is going to have to become a "reasonable" statistician.
These are the pieces of hard won statistical knowledge I wish someone had taught me 20 years ago rather than my meandering, random walk approach.
1. Non parametric statistics. These are statistical tests which make a bare minimum of assumptions of underlying distributions; in biology we are rarely confident that we know the underlying distribution, and hand waving about central limit theorem can only get you so far. Wherever possible you should use a non parameteric test. This is Mann-Whitney (or Wilcoxon if you prefer) for testing "medians" (Medians is in quotes because this is not quite true. They test something which is closely related to the median) of two distributions, Spearman's Rho (rather pearson's r2) for correlation, and the Kruskal test rather than ANOVAs (though if I get this right, you can't in Kruskal do the more sophisticated nested models you can do with ANOVA). Finally, don't forget the rather wonderful Kolmogorov-Smirnov (I always think it sounds like really good vodka) test of whether two sets of observations come from the same distribution. All of these methods have a basic theme of doing things on the rank of items in a distribution, not the actual level. So - if in doubt, do things on the rank of metric, rather than the metric itself.
2. R (or I guess S). R is a cranky, odd statistical language/system with a great scientific plotting package. Its a package written mainly by statisticians for statisticians, and is rather unforgiving the first time you use it. It is defnitely worth persevering. It's basically a combination of excel spreadsheets on steriods (with no data entry. an Rdata frame is really the same logical set as a excel workbook - able to handle millions of points, not 1,000s), a statistical methods compendium (it's usually the case that statistical methods are written first in R, and you can almost guarantee that there are no bugs in the major functions - unlike many other scenarios) and a graphical data exploration tool (in particular lattice and ggplot packages). The syntax is inconsistent, the documentation sometimes wonderful, often awful and the learning curve is like the face of the Eiger. But once you've met p.adjust(), xyplot() and apply(), you can never turn back.
3. The problem of multiple testing, and how to handle it, either with the Expected value, or FDR, and the backstop of many of piece of bioinformatics - large scale permutation. Large scale permutation is sometimes frowned upon by more maths/distribution purists but often is the only way to get a sensible sense of whether something is likely "by chance" (whatever the latter phrase means - it's a very open question) given the complex, hetreogenous data we have. 10 years ago perhaps the lack of large scale compute resources meant this option was less open to people, but these days basically everyone should be working out how to appropriate permute the data to allow a good estimate of "surprisingness" of an observation.
4. The relationship between Pvalue, Effect size, and Sample size - this needs to be drilled into everyone - we're far too trigger happy quoting Pvalues, when we should often be quoting Pvalues and Effect size. Once a Pvalue is significant, it's higher significance is sort of meaningless (or rather it compounds Effect size things with Sample size things, the latter often being about relative frequency). So - if something is significantly correlated/different, then you want to know about how much of an effect this observation has. This is not just about GWAS like statistics - in genomic biology we're all too happy about quoting some small Pvalue not realising that with a million or so points often, even very small deviations will be significant. Quote your r2, Rhos or proportion of variance explained...
5. Linear models and PCA. There is a tendency often to jump to quite complex models - networks, or biologically inspired combinations, when our first instinct should be to crack out the well established lm() (linear model) for prediction and princomp() (PCA) for dimensionality reduction. These are old school techniques - and often if you want to talk about statistical fits one needs to make gaussian assumptions about distributions - but most of the things we do could be either done well in a linear model, and most of the correlation we look at could have been found with a PCA biplot. The fact that these are 1970s bits of statistics doesn't mean they don't work well.
Kudos to Leonid Kruglyak for spotting some minor errors