The new moon, or Noumenia, is the beginning of the lunar month and a time for renewal of the self, of vows and of bonds. It is also a good time to make plans for the coming months. As many Pagans worship according to the lunar calendar, many of Hellenic tradition worship at the New Moon and I am one of these. For each New Moon I honor my household Gods as well as Selene and I hold a special feast for different God or Goddess each month.
The day of the dark moon is Hecate's Supper. This is the day of purification. On this day I perform a cleansing juice fast and I clean the entire house and give extra attention to the altars inside and outside of the house and sweep the whole house with my broom. All of the detritus from the cleaning and sweeping I place on Hecate's Altar at sunset along with a food offering. While cleaning up, I gather together items I know longer need and take them to the Goodwill.
On the day of the new moon, I bring in fresh flowers and spread a clean tablecloth on my table and bake a fresh loaf of bread and cook a fabulous dinner with a special dessert and then I host a Special Dinner Family Ritual or Devotional Rite in honor of the God or Goddess of the occasion. Over dinner, we take some time to review the blessings and troubles of the past month and make plans for the coming month.
When the new moon is high in the sky, I go out alone to honor Selene who is one of my Patrons. This is a fairly simple ritual. I walk barefoot out to the garden (unless it's too cold to tolerate it) and pour some water into her offering bowl. I recite the Homeric Hymn to Selene and then meditate for awhile and do some scrying in the bowl.
The next morning is the Agaithos Daemon, dedicated to the household spirits, including Hestia and our own naughty gremlin whose name it just occurred to me I do not know and the spirits of our land, which seem to manifest as red winged blackbirds, at least in the summer.
While I reserve the New Moon for worship, many Witches will use it as a starting point for drawing magic.