Because people in ancient Greece had only very clumsy ways of writing down numbers, they didn’t like algebra. They found it very hard to write down equations or number problems. Instead, Greek mathematicians were more focused on geometry, and used geometric methods to solve problems that you might use algebra for. Greek mathematicians were also very interested in proving that certain mathematical ideas were true. So they spent a lot of time using geometry to prove that math facts were always true, even though people like the Egyptians and Babylonians already knew that these math facts were true most of the time anyway.
Greek people in general were very interested in rationality, in things making sense and hanging together. They wanted to tie up the loose ends. They liked music, because music followed strict rules to produce beauty. So did architecture, and so did mathematics.
The first famous Greek mathematician was Thales (who was actually from a Greek city in West Asia). In the 600s BC, Thales figured out how to use shadows to calculate the height of the Egyptian pyramids. About a hundred years later, Pythagoras proved that the Pythagorean Theorem was always true. Then in the 300s BC, Euclid (who was born in Egypt, in Africa, but spoke Greek) wrote a famous geometry book proving many more mathematical ideas about the area of a circle, the volume of spheres, and much more.
Archimedes probably knew Euclid; he lived in a Greek city in Sicily. Archimedes worked on getting a more accurate number for pi, and a proof for calculating the circumference of a circle. (So all of these men spoke Greek, but none of them actually lived in Greece.) Probably both Euclid and Archimedes knew something about the work Indian mathematicians were doing at the same time with infinity and combinations. Possibly Indian mathematicians also knew about Euclid and Archimedes’ work.
The Joy of Pi, by David Blatner (1999). It’s not all about ancient Greece, but some of it is. For teenagers.
History of Greek Mathematics: From Aristarchus to Diophantus, by Thomas L. Heath (1921, reprinted 1981). A lot of Euclid, but also describes who the other major Greek mathematicians were and what they did.
Episodes from the Early History of Mathematics, by Asger Aaboe (1997).