how did the “Greek military” work back then?

Most of the time it quite simply didn't. There was never any sort of unified all-Greek, or Pan-Hellenic to use their terms, military. On a broad level, that was purely practical. There were Greek city states dotting the coast from modern Crimea to Spain. If you just look at the Aegean Sea - the part of the Mediterranean between modern Greece and Turkey - even that region never had a united military force. If you get even narrower and just look at the area of modern, mainland Greece its the same outcome.

What the Greeks did have is a long history of alliances that historians usually call "leagues," usually named for a region where many of the local cities got together to aid one another. We don't know a ton about how these alliances first arose. Even though it's not technically considered part of the Greek "Dark Age," the centuries between 800-500 BCE (the Archaic Period of Greek history) are not very well documented. There's some evidence that similar treaties were established among trading partners, and the earliest examples probably started emerging as brief alliances around 800-700 BCE.

The best understood of these early leagues was the Amphyctionic League, formed right around 700. Technically, some scholars use "Amphyctony" to refer to any of these leagues because it just means "a group of neighbors," while others use it to refer to leagues focussed on a common temple. The most famous, and the one meant as The Amphyctionic League one was focused on Delphi - the home of the most important oracle in the Greek world. It's initial form was probably the closest mainland Ancient Greece ever came to a truly Pan-Hellenic alliance when most of the cities in Greece contributed to The First Sacred War (596-585) to stop the city of Kirrha from robbing and threatening pilgrims on their way to Delphi.

This alliance did not unify the mainland Greeks in any lasting way. When the Persian Empire began threatening Greece, cities that had once worked together in the First Sacred War found themselves on opposing sides of the Greco-Persian Wars.

This is the episode most people think of when they talk about a single "Greek military" in ancient history. In 481 BCE, the Persian king, Xerxes, began marching toward Greece at the head of the largest army any Greek had ever seen. As the Persians approached, a new alliance we call the Hellenic League formed to opposed them, but this was not an all Greek army in any sense.

Most of the large, square-ish, southern peninsula called the Peloponnese was allied with, or ruled by, Sparta in a system called the Peloponnesian League. To their east, the smaller region of Attica was largely dominated by Athens, but hosted a few independent cities that Athens and Sparta could put informal pressure on. Together with a few island city states, these cities formed the Hellenic League, as it's called by historians. However, the rest of Greece either elected to stay out of the fight or join the Persians.

Most of the Aegean region was already ruled by the Persians, including all of the Greeks in Libya, Egypt, modern Turkey, and the northeastern part of modern Greece (which was considered Thrace and Macedon in antiquity). As the Persians approached, the kings of Greek cities in Thessaly actively invited Xerxes in, and once they were in Thessaly most of the cities between there and Attica went quietly. There was some independent resistance, but it was crushed.

The most important city to surrender without a fight was Thebes, and after the final Persian defeat in the the battle of Plataea, a siege of Thebes was actually the last action on the mainland under the Hellenic League banner. The Hellenic League carried on with a counter offensive lasting a few more years, but ultimately political differences between Athens and Sparta caused the Hellenic League to split because the Peloponnesian League withdrew its support in 477.

Following that split, Athens and its remaining allies (now including cities recently liberated from Persian rule) formed a new alliance called the Delian League, with a shared treasury on the island of Delos. This was the last real attempt to carry on an alliance of independent Greek city states. The Delian League offered members a choice of providing ships and men for military service or paying a tax/tribute to the shared treasury in order to support other cities who did provide active service. Many cities chose the latter option, while Athens itself always provided the bulk of the military.

Over time, this supposed league of allies became a de facto Athenian Empire, and by 470 the Athenians were keeping their "allies" in the League by force. The island of Naxos was the first of many to try and leave only to be besieged and sacked by the Athenians. Over the following decades, this activity and increasing Athenian economic and military strength repeatedly put them at loggerheads with Sparta. Sparta's Peloponnesian League had always been something of a Spartan kingdom too, but it was actually less domineering than the Delian League by the time they went to war for the first time in 460.

That was the First Peloponnesian War, and through the course of that conflict and the subsequent Second Peloponnesian War, the Greek leagues mostly transitioned from alliances of equally independent cities to semi-imperialist organizations. In turn: Athens, Sparta, and Thebes all had their terms as the head of the dominant leagues in the Greek world, and they functioned far closer to a central government than just one of many allies. Those turns ended because the next dominant league pried enough allies away from their rivals, and had enough Persian support, to win wars and seize power.

After the Corinthian War (395-387), Persian negotiators settled a peace in Greece where none of these leagues was really dominant and they all sort of co-existed because they mostly went back to only coming together in war time. Several leagues, including anew form of the Amphyctionic League tried to challenge Macedonian expansionism in the mid-4th Century BCE, but failed. Phillip II of Macedon technically placed all of Greece in what historians call the Corinthian League, but that was mostly synonymous with the Greek province of a growing Macedonian Empire.