The Greeks weren't originally from Greece. They were originally from around Asia and moved there and settled colonies to spread around Greece. A polis is a city-state in Greece and an acropolis is a settled area usually on a hill for defense purposes. For example lets say the Persians were attacking the Greeks (It actually happened) they would go in the acropolis and hide there for defense while the soldiers fight.
Greek mythology back then was important because it was their religion and it taught about the nature around them and used their gods, goddesses, heroes, etc. as their main characters. An epic is a really long poem, usually used heroes like Hercules, for example the Iliad is about the last few weeks during war in Troy but not all the facts are true but Achilles was a great role model or Alexander the Great and he was a good military leader and conquered many empires. The Odyssey is about Odysseus coming back from the Trojan war and Odysseus said it would only take a few weeks but instead it took 20 years! He had to go through many torturous quests just to get home to his wife, Penelope, and his son.
In Ancient Greece there used to live a man called Archimedes was a person who studied math, physics, engineering, inventor, and astronomer. Archimedes was the one who found the formula of Pi (3.14159) and he defended Syracuse with his amazing inventions.