The art of astrology, the study of the stars in order to predict future events, and to interpret people's characters, was already very advanced before the Greeks began to interest themselves in the stars. The Egyptians and especially the Mesopotamians had done a lot of scientific observations and had named the constellations of stars, and many individual stars.
The Greek contribution to astronomy was not so much in observation as it was in applying logical thinking and geometry to these observations. That is how Greek scientists figured out that the earth went around the sun, calculated the size of the earth, and understood that the moon went around the earth. Some famous Greek astronomers were Thales, who figured out that the earth was round, Anaxagoras, who figured out what caused eclipses, and Aristarchus, who figured out that the earth went around the sun.
But at this point Greek astronomy ran into a big problem. Many astronomers pointed out that if Aristarchus was right, and the earth went around the sun, then the earth was also moving relative to the stars. If so, you should be able to observe parallax when you looked at the stars. Or, the stars had to be trillions of miles away, so far that you couldn't observe the parallax. When Greek astronomers like Hipparchus looked at the stars, they couldn't see any parallax. Nobody can see it, with just their eyes.
No parallax meant one of two things. Either the sun was really going around the earth, or that the stars were so far away that you couldn't observe the parallax. It seemed unlikely that anything could be trillions of miles away from anything else, so Greek astronomers went back to thinking that the sun went around the earth. But they kept looking for that parallax. A thousand years later, when telescopes were invented, the first thing astronomers did was to look for that parallax - and Galileo still couldn't see it, but he did see the moons of Jupiter going around Jupiter, and that convinced a lot of people that the earth really did go around the sun.
To find out more about Greek astronomy, check out these books from Amazon.com or from your library:
The Librarian Who Measured the Earth, by Kathryn Lasky (1994). An account of the life and work of Eratosthenes, who figured out the circumference of the earth. Explains how he did it. For kids.
The Shining Stars: Greek Legends of the Zodiac, by Ghislaine Vautier, Kenneth McLeish and Jacqueline Bezencon (reprinted 1989). A Greek myth for each star sign, with drawings of the constellations so you can find them in the sky. For kids.
Greek Astronomy, by Thomas Heath (1932). A collection of what ancient Greek writers had to say about astronomy, in their own words, with a long introduction. For adults.
The History & Practice of Ancient Astronomy, by James Evans (1998). Includes both the history, and directions to actually re-do the experiments that ancient Greek astronomers used to figure out their conclusions. For adults.
Greek Science After Aristotle, by G. E. R. Lloyd (1975).