What courses / subjects are most important to the field of Data Science?
Probably the most useful (and fun) answer is provided by the excellent Swami Chandrasekaran ... in the form of a subway map:



I would add just one comment to that:  school (or learning for the sake of learning) doesn't provide the same kind of experience as solving a real problem does.  Therefore, to learn, find real problems you can tackle.  To stay motivated, make them problems you really care about.  Kaggle competitions are a great place to start.  Even reproducing a winning solution (perhaps with some variations, or on a new dataset) will be a huge learning experience.