The New Moon: "My Faithful Witness"
We are told in Genesis 1:14 that the moon (among other heavenly bodies) was created for "signs, seasons, days, and years." We therefore need to watch them and keep track of what they are doing--but never be tempted to worship them. (Deut. 4:19) Psalm 104:19 is even more specific: "He created the moon for Mo'adim [appointed times]". That is because it is what determines the dates of the prescribed holy days, which all (except Shavuoth) fall or begin on a particular day of the month. During the flood account we are told that there were 150 days from 17th of the 2nd month to the 17th of the 7th month, showing that at that time a month was exactly 30 days. (Gen. 7:11, 24; 8:3,4) That has changed since then as a result of changes to the earth's orbit caused by external factors, so we have to watch carefully; it is no longer so easy to predict. There are very few actual commands in the Torah relating to the New Moon.
The two silver trumpets are to be blown (B'Midbar/Numbers 10:10) Psalm 81 alludes to blowing it at a particular new moon which is also a feast day (khag)--and there is only one: Yom T'ruah (The Day of the Awakening Blast).
Particular offerings are to be brought: two young bulls, one ram, and seven lambs without defect. (B'Midbar 28:11) Y'hezq'el 45:17; 46:6 say that in the time of the third Temple, an additional offering will be brought to the Temple at New Moon by the prince (possibly the Messiah himself): a young bull, six lambs, and a ram.
During the Messianic Kingdom, it is prophesied that people from all nations will come up to the Temple "from one new moon to another and from one Sabbath to another". (Yeshayahu 66:23) We therefore know it is YHWH's will that this be done, so it is an implicit command.
Traditions
At least by the time of King Sha'ul, the new moon was celebrated for two days with a special feast. (1 Shmu'el 20:18-24) The festival was the day after the moon appeared, largely because much time was required to prepare a feast that included slaughtered animals.
During the divided Kingdom, at least in the Northern Kingdom, it was customary to visit a prophet on the new moon (2 Kings 4:23).
From the time the Temple was built, the Levites offered the slaughter on this day (1 Chron. 23:31) Shlomoh said he built the Temple in part specifically for the purpose of bringing the new moon offerings (2 Chron. 2:4) as Moshe commanded (2 Chron. 31;3), though the particulars are not actually specified in the Torah. There was a tamid ("daily") ascending offering made on the new moon just as on every other day. (Ezra 3:5; Neh. 10:33)
By Second Temple times, according to the Mishnah, when the moon was sighted, the initial witnesses had to appear before a court in Yerushalayim and testify of what they had seen. They were interrogated with questions regarding which direction the points of the sliver moon were facing, to be sure they had actually seen the moon when visibility was very low. When the Sanhedrin was satisfied that they were bearing a true witness, and two testimonies coincided, they would say, often repeatedly, "It is set apart", and a large fire was lit on the Mount of Olives. It was kindled using cedar sticks, reeds, oleaster wood, and flax tow.
When watchmen posted at Sarteba, on the edge of the Jordan River Valley north of Y'rikho, would see it, they would relay it to Agrippina, Hauran, Beyt Baltin, and possibly other points so that the word could get out all over the Land that it was a new month. They would wave the flaming brands until they saw their fellows on the next hill, then the third hill do the same. The Samaritans sometimes spitefully built fires to give a false report and confuse the Jews, so other means of communication began to be devised.
In the months when it matters which day it is (when there is a set time commanded by YHWH that month), messengers were sent out: in the first month (because of Pesakh), the fifth month (because of Tisha b'Av), the sixth (because the new year was coming), the seventh (because of the determination of the set feasts), the ninth (because of Hanukkah), and the twelfth (because of Purim). They went as far as Hamath in Syria, and they ruled that they could violate the Sabbath to do so, because of the importance of determining when the offerings would be, based on the Torah command to proclaim the feasts in their appointed seasons. (Lev. 23:4) If one had to wait an extra day to proclaim Yom T'ruah (Rosh haShanah), the appointed time would already be past.
Certain people were considered invalid to serve as witnesses (on the same level as women): Those who play with dice, those who lend at interest, those who race pigeons, those who trade in the produce of the seventh year, and slaves. These people were considered susceptible to bribery (i.e., the Samaritans might offer them money to give a false report).
In modern Judaism, New Moon celebrations are not widely practiced. They are often limited to women--but probably not for the reason you'd think.
There are traditions in Native American cultures that say all women used to start bleeding right before the New Moon, and they had taboos that seem like they might well have stemmed from the Scriptural niddah laws (regarding the time a woman is "off-limits" to her husband each month). Women are not allowed in sweat lodges and must stand outside the circles in other ceremonies when they are in their moon time. They were believed to have great power at that time since they could bleed so much and yet not die as a warrior who lost as much blood might, and the men thought that power would interfere with the hunt or take away the power of the medicine bundles. They would all actually go to a separate lodge during that time, leaving the grandmothers and fathers to care for the children. (Marcia Stack)
There may be some carry-over from a similar idea in Judaism, because it does remain a custom in some Jewish communities for women to refrain from work on Rosh Khodesh. But a midrash (story from the Talmud) gives a different reason. It says that each of the new moons was originally intended to represent one of the twelve tribes of Israel, just as the three major festivals each represent one of the three patriarchs. However, because of the sin of the golden calf, the holiday was taken away from the men and given to women, as a reward for the women's refusal to participate in its construction. How do we know the women didn't participate? Because Exodus 32 says that "the people" asked Aharon to make an idol. Aharon told them to get the golden rings from their wives and their sons and daughters. But the biblical verse says nothing about "the people" getting the rings from their husbands, only from wives and sons and daughters, so we can infer that "the people" it was speaking of were the men. They "broke off the golden rings that were in their ears". I.e., the same people gave their own jewelry. The midrash explains that the men went back to their wives and the wives refused to give their gold for the creation of an idol. As a reward for this, the women were given the holiday that had been intended to represent the tribes. (Judaism 101)
There are a variety of ways of marking the new moon today, and many of them with valid reasons behind them. The Apostle Paul said we should let no one but the Body of Messiah be our judge in relation to the new moon. (Col. 2:16)
The Sabbath before Rosh Khodesh is known as Shabbat Mevarekhim, which means "the Sabbath of blessing." After the Torah reading in the Shabbat service, the prayer leader holds the Torah scroll, recites a blessing for the month, then announces the day of the upcoming week when the new month will begin and the name of the new month. Shabbat Mevarekhim is not observed during the month of Elul to announce the beginning of the month of Tishri, the month in which the fall festivals occur, since, according to Hasidic tradition, YHWH himself blesses the first of Tishri, the anniversary of Creation, while giving the privilege of blessing the rest of the months to the people of Israel. (From Judaism 101)
The day before the new moon has sometimes been treated as a "yom kippur katan" (yoam kee-POOR ka-TAHN)--a small Yom Kippur, a time to pause and take a hard look at where our relationship with YHWH stands, and repent where needed, to be ready for the renewal.
What themes are associated with the New Moon?
Renewal: The literal meaning of Rosh Khodesh is "head of the renewing". Only by extension does the term mean "month" as meaning the period between the two new moons. (But it is used this way in Gen. 29:14 and Num. 11:20-21.)
Rebirth: Another reason the new moon has often been associated with women is that the conjunction of sun, moon, and earth (in that order, which is why the moon is dark just before new moon) is called molad ("mo-LAHD")--the birth. Yahshua spoke to Nicodemus of new birth at night (Yochanan 3)--certainly referring to the new moon cycle, which would be quite visible then. Victor Hillel Reinstein writes, "At the moment of the conjunction, the end and the beginning are as one. It is as harmonious a moment as there can be. This regularly recurring end and beginning in the same moment reflects the continuity of an eternal recurrent cycle. In the harmony of the molad is an intimation of [YHWH] as the One without beginning and without end."
Waxing toward fullness--a picture of Israel at its finest. The most joyful festivals (Unleavened Bread and Sukkoth) begin on the fifteenth of the month. We go through times of darkness, but the moon reminds us that the full light will always come back again. (Compare Matthew 13:43 with Daniel 12:3 and Yeshayahu/Isaiah 30:26.)
Rejoicing (Hoshea 2:11). For this reason, one is traditionally forbidden to fast or express other symbols of mourning on the new moon, just as on the Sabbath.
Rev.22:2 tells us that the Tree of Life bears fruit every month (or new moon) and its leaves are for the healing of the nations (which could just as well be translated "tribes", so we should take it both ways. It is an appropriate time to pray for healing where needed.
Continuity: YHWH (and apparently the Messianic king too) will be honored and peace will remain until the moon no longer exists. (Psalm 72:5-7)
In parallel poetry, the moon is called a "faithful witness in heaven." (Psalm 89:37)-- a name also applied to Yahshua (Rev. 1:5) So there must be an implied connection.
Another word for the moon itself is yareaH, which is often used interchangeably with Khodesh. (1 Kings 6:1; 8:2) It is also called levanah (the "white one", a feminine form, which suggests a bride; Jacob understood the moon in Joseph's dream to be speaking of his mother in Genesis 37).