John has some suggestions below for a fairly serious effort to learn how to use QM in practice. I can think of some more introductory material to get you started. That would include one of the lectures in Feynman's "Character of Physical Law", a little book you're sure to enjoy. There's also volume 3 of Feynman's Lectures on Physics. Thomas Moore's "Unit Q" is a another nice beginning intro, intended for undergrads. Mike W.
"Understanding" quantum mechanics is quite a different task than understanding the language in which most quantum mechanics is written! I'm assuming the latter is your true question.
At the very least you need a basic understanding of linear algebra, multivariate calculus and differential equations (up to a rudimentary understanding of partial differential equations). If you're really serious about understanding the mathematics I would either purchase or borrow or acquire by some other means a copy of "Introduction to Quantum Mechanics" by David Griffiths. Griffiths is an excellent writer of undergraduate-level physics textbooks and his introductory text on QM is no exception. It will give you a sense of the mathematics behind basic quantum mechanics. It includes a brief review of linear algebra as an appendix. It's helpful but not most likely not an adequate substitution for a linear algebra class.
More advanced understanding branches, understandably, into more advanced mathematics. However, a thorough understanding of those three subjects will take you far.
3 or 4 semester-long mathematics classes and probably 4 semester-long physics classes (covering basic physics and, later, Griffiths) will give you a pretty decent understanding of basic quantum mechanics. That's anywhere from 2-4 years. If you're clever and able to teach yourself, the time range can be anything from days (if you're extremely clever) to months to years to decades.
(published on 07/09/2011)