In my humble opinion, all biologists training today should develop some basic computational proficiency. If your graduate program does not have a formal framework for this, here are some personal suggestions of where to start. Disclaimer: As always, one absolutely must understand the limitations and biases of specific data sources and analysis methods before drawing conclusions from them.
Everything listed is free and open source software unless otherwise noted.
Sometimes, quick operations on Galaxy or a visit to the UCSC Genome Browser or Ensembl are sufficient. When it's not, you'll want a repertoire of other tools to do the job straight away. The more you know, the more likely you are to get a job done in short order.
While you can use many of the tools listed below on Windows (Python, R, Zotero, LyX, etc.), I'd strongly advise you learn to use the Linux command line (you'll be glad you did when you need to run computationally intensive tasks on a server or computational cluster). I'd suggest messing around with Linux on your local computer. Ubuntu is a relatively easy to learn Linux distribution for newcomers and can ease the transition for someone who's never used anything but Windows. Alternatively, if you have a Mac, you can learn a lot using the Terminal.
Of course, you'll also want to learn to use various tools to work on another computer (a "server") from your local machine such as ssh (opening a command line session on the server), scp (transferring files), and vim or emacs (for text editing). This list is very incomplete, but it should get you started and Google can help with the rest.
If you're a wet lab biologist, chances are you won't need a heavy programming language like C, but you should definitely learn to script (to automate boring, mindless tasks). If you only have time for one scripting language, I'd suggest Python. For trivial tasks (for example, reformatting files), you can save yourself a lot of time by also learning shell scripting.
If your analysis involves doing a large set of complicated tasks several times, and you can write Python scripts for each step, I'd suggest automating them with ruffus.
R is an unparalleled environment for statistical computation and visualization, with tons of existing libraries for routine bioinformatic analyses.
Forget Endnote. Zotero, a Firefox plugin, leaves it in the dust in every way.