Many, perhaps even most, pre-historic civilizations made observations of the sky and tracked motions of the sun, moon, and planets. There is a 27,000 year old mammoth tusk found in the Ukraine which contains markings which may be a count of lunar cycles. Could these people be considered the earliest astronomers, making observations and noting patterns in celestial movements? As far back as 1600 BC, the Babylonians had kept detailed written observations, compiled star catalogs, and were undertaking long-term studies of planetary motions and eclipses.
However, most of these early cultures integrated their observations intimately with their spiritual beliefs, and their "astronomers" were more frequently religious shamans or priests, responsible for predicting astronomical events, integrating them into the mythology of the group, and assuring any astronomically-caused affects on the society would be benign. And, most importantly, they used religion to explain the physical causes of celestial motions, so had no need to develop other models.
By 300 BC, the Greeks had managed to amass the astronomical knowledge of the Babylonian, Egyptian, and other cultures. But the early Greek astronomers did what the Babylonian astronomers would never have dreamed of -- they devised geometrical, physical models for the movements of the sky. Although they still had their mythology, the Greeks added curiosity, which drove them toward trying to understand reality. They moved toward a true scientific method -- to observe and thence explain what is seen, and predict observations accurately from those models. Aristotle, and Ptolemy, heavily influenced by Plato's teachings, devised a system of uniform and perfect motion which could, sort of, explain and predict the movements of the cosmos. They, of course, assumed the Earth was the center. Unfortunately, they were dead wrong.
Curiously enough, a Greek scientist named Aristarchus did propose a sun-centered system. However, when the great library at Alexandria was burned, all the major writings of Aristarchus, and indeed of the rest of the great scientists of the time, were destroyed. With the loss of the Alexandria library began a period of European dark ages, where, thanks to the philosophies and influence of Plato, observations were considered to be distorted versions of reality and only pure thought could produce accurate results. So science and knowledge were suppressed, or relegated to the confines of a social and religious elite. Not until Copernicus, in the late 1400s, was an intellectual revolution to be launched, a revolution which marked the first major shift in our concept of the Earth's place in the cosmos. Ultimately, ableit slowly, this shift shaped modern views of the sun, the solar system, and the cosmos.