Who was the first to systematically explore the study of public speaking?

Aristotle

Depending on who you ask you may get a different answer. In the beginning there were Sophists who were traveling tutors. They educated lawyers and politicians who were interested in persuasive speaking. Plato denounced the Sophists because they relied on persuasion through emotions and little to no truth. Plato and Aristotle both believed in the ethical truth telling. However, Plato relied mostly on truth and logic. Aristotle was the one to step up and introduce the idea of his proofs: ethos (creditability), pathos (emotion), and logos (logic). This was seen as both ethical and effective during his time. Since then philosophers like Cicero and many others have modified Aristotle's version of rhetoric. However, Aristotle's principles are the most widely accepted and taught until this day.