The Torah lists Rosh Chodesh together with all the other holidays on which we offer additional sacrifices (Musafim) in honor of the day’s sanctity. Chazal derive from a verse in Eichah(1:15) that Rosh Chodesh is also called mo’ed (an appointed time) like all the other holidays (Pesachim 77a). In Temple times, they used to blow trumpets on Rosh Chodesh, as it says, “On the day of your gladness, and on your appointed times, and on the beginnings of your months, you shall sound the trumpets over your burnt offerings and over the sacrifices of your peace offerings”(Bamidbar, 10:10).
Because of the sanctity of Rosh Chodesh, a custom developed to go out and greet one’s Rabbi, similar to the custom on the Sabbath to say “Shabbat Shalom” to the Rabbi of one’s shul. ( Melachim 2, 4:23, Rosh HaShanah 16b, Bi’ur Halachah 301:4). There is also a custom to prepare festive meals on Rosh Chodesh(cf. Shmuel 1, ch. 20).
A wonderful expression is used by the Torah when describing the goat that was sacrificed on Rosh Chodesh, calling it “a sin offering for the Lord”(Bamidbar, 28:15). The Talmud (Chulin 60b) explains that in the beginning, God created two great luminaries in the sky, the sun and the moon. The moon, however, made a claim to the Master of the World: “How can two kings share the same crown?” hoping that HaShem would diminish the sun so that it, the moon, could reign supreme. God, however, said to the moon, “Go and reduce yourself.” It replied, “Because I made a justified claim before You, I should reduce myself?” God consoled the moon by saying that the Jews would reckon the months according to its cycle, and that the righteous would be called by its name. But the moon was not consoled, so the Holy One Blessed Be He said, “Bring an atonement for Me, for I diminished the moon.” This is why it says, “And one kid of the goats for a sin offering for the Lord.“.
This contains a very profound concept. On a simple level, the moon’s reduction symbolizes the deficiencies that exist in creation, including the descent that the soul undergoes when it arrives in this world, and all the failures that man experiences during his lifetime. All these failures and deficiencies are prerequisites for subsequent growth, because coping with hardships helps one reach higher heights in the end, as R. Abahu says, “The purely righteous cannot stand where penitents stand” (Berachot 34b). In the meantime, people commit sins which cause great pain in the world. So, in order to relieve the pain and repair the flaws, HaKadosh Baruch Hu commanded us to sacrifice a goat as a sin offering. This is the purpose of Rosh Chodesh, to show us how a new beginning sprouts from the moon’s reduction, which happened as a result of sin and indictment. Therefore, Rosh Chodesh is a good time for repentance, new beginnings, and profound joy. However, until the world is redeemed from all its deficiencies, the joy of Rosh Chodesh is not completely revealed (see also below, sec. 15-16).
 A goat was brought as a sin offering on all the holidays, but no where else does it say, “a sin offering for the Lord.” The other musaf (additional) sacrifices offered on Rosh Chodesh were two bullocks, one ram, and seven sheep as a wholly-burnt offering (Bamidbar, 28:11).