Do things, ask questions, meet people. Sooner or later you will begin to figure the place out. Not long ago I attended a conference on diversity. Discussion centered on expanding the curriculum to include multicultural issues. There was much talk of power structures and how presidents and faculty senates decide what's taught.
Senates have little or nothing to do with what actually gets taught, and Presidents even less.
Departments, of course, most immediately. But more broadly, committees, specifically committees such as the General Education Committee, the Undergraduate Council, the Graduate Council, and Senate committees. And all are appointed or recommended by the Senate Executive Committee. Curiously, no one at the conference seemed to appreciate this. There's more to affecting the direction of the university than simply being on a committee, though.