Start with the basics. Luckily, since there are so many good quality resources available online, there is no need to buy any books. The Ludwig von Mises Institute is a major research institution for Austrian economics, and provides free online access to literally thousands of books, articles and lectures. It has published a plain-English introduction to Austrian economics: "Economics for Real People." When researching online, be careful that you are getting a balanced view of the Austrian school, and that the material does not concentrate merely on one thinker. It should make references to numerous academics, including Mises, Hayek and Rothbard.
Take your studies further by reading a more in-depth introduction, such as the free primer available on the Adam Smith Institute website. The Mises Institute also offers a series of lectures available for free online, intended as a home study course. It also provides a list of required readings for those attending classes at the Mises University; this will give an overview of classic texts in the field. Explore various opinions on Austrian economics, which will give you an idea of the difference between Austrian economics and neoclassical economics.
Attempt to understand contemporary economic events in light of Austrian theories. Blogs such as ThinkMarkets by the Colloquium on Market Institutions and Economic Processes offer commentaries on policy issues such as bank bailouts and fiscal stimulus packages, taking a free-market approach.
Enroll in a course in Austrian economics at a university or summer school. Undergraduate courses in economics at George Mason University and Walsh College of Accountancy and Business Administration have a strong Austrian component, and enable students to specialize in this and related areas. New York University has a masters program in the subject. The Mises Institute also offers some courses, summer schools, scholarships and research opportunities.