See here: https://en.wikipedia.org/wiki/Methodenstreit
The most important Austrians you might want to study are:
- Carl Menger (Microeconomics, Methodology)
- Eugen von Böhm-Bawerk (Capital Theory, Famous for Marxist critics)
- Friedrich von Wieser (Oppertunity cost)
- Ludwig von Mises (Methodology, Buissess Cycle Theory, Political Philosophy)
- F.A. Hayek (Buissess Cycle Theory, Information Economics, Political Philosophy)
- Joseph Schumpeter (Austrian educated but went his own way in many ways)
- Fritz Machlup (Buissess Cycle Theory, Trade)
- Israel Kirzner, not actual Austrian (Entrepreneurship)
- Murry Rothbard, also not Austrian (Political Philosophy)
Of these Hayek, Schumpeter, Machlup and Mises all eventually tought in the US and Kirzner and Murry are US studends of them.
There are books on the topic:
- The Austrian School of Economics: A History of Its Ideas, Ambassadors, & Institutions
Today there is split between those influence by Murray Rothbard, who has a perticular interpretation of Mises and very extrem views in many ways. The 'Ludwig von Mises Institute' is pushing this view, they are partly a lobby organisation. I don't think much of them. They tend to dislike Hayek.
There is the more traditional more academic branch, tody mostly represented by George Mason University.