Mental health warning: this is long, complex, and confusing! Do not proceed if you value your sanity.
Let's start with year numbering.
There wasn't any. In Athens each year was named in honor of the city Mayor-cum-CEO who held office that year, called the Eponymous Archon, meaning archon who names [the year]. Eponymous is a word in English to this day, btw, and its meaning hasn't changed much in 2,500 years. So my first book, The Pericles Commission, opens in the Year of Conon, which we call 461BC.
Named years have caused historians to put a lot of effort into working out the correct and complete archon list. Without it, you can't work out what happened when. It's all very well for X to happen in the Year of Fred and Y to happen in the year of Jane, but which came first?
It gets worse, because every city named their years after their own officials, and no one ever kept a list of corresponding years between cities. The cities also kept different months, as we're about to see, and began their years on different days.
So unless two events happened within shouting distance of each other, no one could ever know which happened first. This is reflected in the words of Thucydides, Herodotus et al., who frequently use terms like, "X happened at about the same time as Y." They're not being deliberately vague; they genuinely have no idea which happened first or how much time separated. They can easily be out by many months. In fact, to compare two distant events, the smallest safe unit of resolution is probably the season.
Greek months are lunar. Every Greek month starts with the sighting of the next new moon. In fact, they called the first day of each month noumenia which means (surprise!) new moon. Every noumenia was considered a particularly holy day. Officials went out of their way to never schedule anything for a noumenia. So if you ever catch my characters attending an official function on a noumenia you can validly beat me up. I did, in fact, in my short story The Pasion Contract have a contract killing arranged for noumenia, but I figure that's acceptable since hired thugs probably aren't all that pious.
There are 29 days, 12 hours, 44 minutes and 3 seconds in an astronomical lunar month. So Greek months were either 29 or 30 days long. Since the new month was defined as starting on the sight of the new moon, there was never much doubt when a new month began.
The definition of "day" was a bit odd. For the Greeks, the old day ended and the new one began at dusk. This makes perfect sense for people working to a lunar calendar. Their day begins with the moon telling them what day of the month it is. But it creates a terminology problem for me. Nicolaos could say at midday, "I'll meet you early tomorrow," and mean that night, leaving you the reader totally confused. I solved this problem by completely ignoring it and sticking to modern convention.
The Athenian months in order were:
No, I can never remember them either (and can barely pronounce them!) but the exotic names are great for atmosphere. I can say things like, "He died on the noumenia of Hekatombion," as if it made sense.
Every city had a different set of names for their months. Just to make it more fun, they started their years at different times too. All the Hellene cities fell (broadly) into two very ancient tribal lineages: the Dorians and the Ionians. I have a feeling, which I've never been able to confirm, that it was mostly Dorian cities which started their year at winter solstice, and mostly Ionian starting at summer. But don't quote me on that. The Athenians, as good Ionians, began theirs on the first new moon after the summer solstice, putting them as out of synch with the modern calendar as you can get.
The lunar month does not divide evenly into 365.2423 days. (Nor does any other sensible number, for that matter. When God created the universe, he really screwed up on this point big time.)
The Greeks tried to fit a lunar calendar into a solar year, so that the months had a fighting chance of being consistent with the seasons. Such calendars are called lunisolar.
Twelve lunar months fall short of a solar year by about 11 days. The Athenians handled this by inventing an extra month every two years. And not just inventing, but duplicating the previous month. It is said, but I've never seen the proof, that the month they usually duplicated was Poseidon, so every second year there'd be Poseidon I followed by Poseidon II. Or not, if they decided to duplicate a different month instead. The guy who got to decide was the Eponymous Archon.
This still didn't fix the problem completely since twice 11 does not equal 29. Not to worry...the archons invented extra days at random to pad things out. This was a useful trick for buying extra rehearsal time before important festivals if people weren't ready. I'm not kidding...this actually happened! Once, the calendar was frozen for 4 days before a Great Dionysia.
Since the noumenia of Hekatombion is the first new moon after the summer solstice, it acts like a reboot for the calendar. No matter how screwed up the calendar had become, there was always a clean reboot in the future.
They couldn't even manage to number the days in a month consecutively. Every month was broken into three sections: moon waxing, full, and waning. After noumenia, the next day was called 2nd Waxing, then 3rd Waxing, and so on to 10th Waxing. Then the system changes to 11th, 12th, 13th...19th, and then earlier 10th! 10 doesn't normally come after 19, but that's how it worked! The earlier is very important because the following day was later 10th. Yes, they had two 10ths in a row: earlier and later. After later 10th it counted down: 9th Waning, 8th Waning, ..., 2nd Waning, and ending with hena kai nea, meaning old and new. On a 29 day month 2nd Waning was dropped.
If you're getting the impression the Greek calendar was insanely chaotic, you're right. A modern astronomer confronted with the Greek calendar would probably be driven to self-harm, or more sensibly, Greco-harm.
The only thing common across every calendar in every city was...the Olympics. They happened every 4 years at a more or less consistent time, fixed by the leaders of the city of Elis, who were always the hosts. That's why some ancient authors might refer to an event as happening in the year of the 74th Olympiad, or whatever. It's their attempt to make dating sensible across all of Hellas, but that's the closest they ever got to unity.