The Hellenic Calendar
The first thing to remember is that there was no one, single, unifying calendar in Ancient Greece. Different regions had their own ways of reckoning the days and festival dates would vary widely. Even different villages within the same region might have variances on their calendars and unless there was a major festival coming up it wasn't rigidly observed. Secondly, there was also no one, single, unifying Greek religion. Ancient Greek religion was cult-based, with various cults operating out of the same city and being more or less prevalent depending on the location. For example the urbane Athens would have been more concerned with civic cults whereas countryside locations, such as Eleusis, would support more rural cults. Because of this some festivals celebrated in one region would be ignored, or greatly changed, in another.
The most complete calendar we have however does come from Athens and the Attic Region, and this is the one most often utilised and adapted by modern Hellenists.
Reckoning the Months
The Athenian Calendar has twelve lunar months, with an intercalculary month repeated every few years or so, just to keep the dates fairly regular. The first day of the month began on the day after the new moon, usually when the new lunar crescent was first seen in the sky. The day of the new moon was the last day of the month. The first day of the New Year started on the sighting of the new lunar crescent, after the Summer Solstice. The days began on the evening of the day before.
Some Hellenists do count their days starting at sundown of the day before, however others (including myself) disregard this unless there is an important festival. So, looking at the above "formula", 2014 (in the Northern Hemisphere) would look something like this;
1st January: New Moon, last day of the month of Poseidon, Hekate's Deipnon
2nd January: Noumenia, first day of the month of Gamelion
16th January: Full Moon
30th January: New Moon, last day of the month of Gamelion, Hekate's Deipnon
1st February: Noumenia, first day of the month of Anthesterion
14th February: Full Moon
1st March: New Moon, last day of the month of Anthesterion, Hekate's Deipnon
2nd March: Noumenia, first day of the month of Elaphebolion
16th March: Full Moon
30th March: New Moon, last day of the month of Elaphebolion, Hekate's Deipnon
1st April: Noumenia, first day of the month of Mounikhion
15th April: Full Moon
29th April: New Moon, last day of the month of Mounikhion, Hekate's Deipnon
30th April: Noumenia, first day of the month of Thargelion
14th May: Full Moon
28th May: New Moon, last day of the month of Thargelion, Hekate's Deipnon
29th May: Noumenia, first day of the month of Skirophorion
13th June: Full Moon
27th June: New Moon, last day of the month of Skirophorion, Hekate's Deipnon
28th June: Noumenia, first day of the month of Hekatombaion, also the start of the Athenian New Year
12th July: Full Moon
26th July: New Moon, last day of the month of Hekatombaion, Hekate's Deipnon
27th July: Noumenia, first day of the month of Metageitnion
10th August: Full Moon
25th August: New Moon, last day of the month of Metageitnion, Hekate's Deipnon
26th August: Noumenia, first day of the month of Boidromion
9th September: Full Moon
24th September: New Moon, last day of the month of Boidromion, Hekate's Deipnon
25th September: Noumenia, first day of the month of Puanepsion
8th October: Full Moon
23rd October: New Moon, last day of the month of Puanepsion, Hekate's Deipnon
24th October: Noumenia, first day of the month of Maimakterion
6th November: Full Moon
22nd November: New Moon, last day of the month of Maimakterion, Hekate's Deipnon
23rd November: Noumenia, first day of the month of Poseidon
6th December: Full Moon
22nd December: New Moon, last day of the month of Poseidon, Hekate's Deipnon
23rd December: Noumenia, first day of the month of Gamelion
Note that the Athenians didn't mark the occasion of the full moon, I just added them to divide the months up.
The first day of the month is known as the Noumenia, and always follows the new moon. As the lunar crescent isn't always visible on the day immediately following the new moon, sometimes the day after Noumenia is called "Noumenia Kala Selene", indicating this is the day of the actual visible new moon. However the Noumenia is always considered the first day of the month.
The Noumenia was a monthly religious observance and sacred to a number of deities, especially those governing the household. Hestia, Apollon Noumenios, Zeus Ktesios, Hekate, Hermes and Selene were honoured at family altars, as well as personal daimons (spirits) and ancestors.
On the last day of the month, the new moon, the goddess Hekate was honoured with a deipnon (supper). This would usually consist of a meal made up from any of the following; bread, cake, cheese, fish, garlic, onions, leeks, eggs and wine. This was left at a crossroads, burial ground or other liminal location. None of the food was shared with the worshipper, the entire meal being given to Hekate and the restless dead. Sometimes household sweepings were also offered to the goddess, as the whole house was cleaned on the new moon in preparation for the Noumenia the following day. The purpose of the deipnon was to evoke Hekate's protection on the home as she is an apotropaic goddess who wards off negative influences.
Monthly Holy Days
In his Theogony, Hesiod states that there are also a number of days each month that are sacred to particular deities, spirits and heroes. After Noumenia, which is the first day of the month, follows these various holy days;
2nd: Agathos Daimon
4th: Aphrodite, Hermes, Herakles, and Eros
The last three days: All khthonic (Underworld) deities and spirits
The last day (new moon): Hekate
The Attic Calendar on Wikipedia: http://en.wikipedia.org/wiki/Attic_calendar
Living by Moon and Sun on A Forest Door: http://forestdoor.wordpress.com/2010/08/10/living-by-moon-and-sun/
The HMPEA Hellenic Calendar. The link directs to the month of Boedromion which starts on Friday the 6th of September: http://www.numachi.com/~ccount/hmepa/calendars/698.1.Boedromion.html