I'm starting my mater's thesis in computer science. I'm interested in computational Origami (algorithms mainly); I've read a little about the subject and I'm worried because I lack some of the knowledge that I think is necessary to do research in that area. But still I'd like to start and want to ask: which is better an approach: learn the necessary background as I go and wheneve a necessity pops up or learn it first and then start. I have 4 to 5 months to finish my thesis.
In my experience, it is better to learn stuff as you need it. Otherwise, you don't have the motivation to really "get" it, and there is always the risk of getting lost in the forest of interesting stuff out there (that in the end is of no use). Be at the lookout for results that could help you (Google or citeseerx or ..., or even this site, are your friends).
Another piece of advise: Start writing right now. Editing your ramblings into something coherent later on is easy, sitting in front of a blank screen with writer's block when the deadline approaches is extremely painful. Write down what you do right now; if it works out it stays for later, if not it gets erased (and you don't get embarassed by your dumb attempts ;-). That way it doesn't happen to you that you know something is right, but don't remember why... and have to redo lots of work (happened to me!).
Make a short (3-4 line) summary of each paper or other reference that looks relevant, add it to your bibliography. Reading the stuff so you are able to extract what is relevant to you helps understanding the stuff, the summaries help in locating material later. And the collection of summaries will be your chapter on state of the art (after cleanup).
As a final comment, use LaTeX (it looks much nicer, has very good handling of bibliographies ;-), and perhaps asymptote for figures (it is a bit hard to learn, but has C++-like syntax to manipulate points and define lines, and produces excelent results). Put the whole stuff under version control (my favorite is git, but use whatever you are confortable with). That way you'll be able to have an up-to-date backup (preferrably on some other machine!), and go back if you really mess up (Murphy's lat assures us you will).
I agree that you will be more motivated for deep understanding once you have already begun working. At the same time, you want to make sure that you have covered enough background that you don't "reinvent the wheel". I suggest alternating your background reading and more focused work. Once you've started working, it will help focus your reading. Your reading may in turn spark new ideas for your work.
I find it helpful to read first for breadth, and then for depth. A good survey paper can save a lot of time here, but if not, I like to take a very brief look at a whole range of papers to determine what's important, then take a deeper look at the papers which have had the most impact or are most important to my particular interest. I read just the introductions of a few of the most recent papers. It helps orient me to what have been the seminal papers over time. Then I take a look at those papers. Going in historical order can help, since later papers build on earlier papers, but also be aware that later papers may refine or simplify earlier ideas as well.