Colossians 2:16. So don’t let anyone criticize you for what you eat or drink, or for not celebrating Jewish holidays and feasts or new moon ceremonies or Sabbaths. For these were only temporary rules that ended when Christ came. They were only shadows of the real thing—of Christ himself. 

Isaiah 66:23 And it shall come to pass, that from one new moon to another, and from one Sabbath to another, shall all flesh come to worship before me, says the LORD. 

Who Celebrated the new moon festivals in the bible: 

1 Samuel 20:5 And David replied, "Tomorrow is the beginning of the celebration of the new moon. Always before, I’ve been with your father for this occasion, but tomorrow I’ll hide in the field and stay there until the evening of the third day. 

2 Chronicles 2:4. "I am about to build a temple for the Lord my God," Solomon told Hiram. "It will be a place where I can burn incense and sweet spices before God, and display the special sacrificial bread, and sacrifice burnt offerings each morning and evening, and on the Sabbaths, and at the new moon celebration and other regular festivals of the Lord our God. For God wants Israel always to celebrate these special occasions. 

Nehemiah 10:32. We also agreed to charge ourselves annually with a Temple tax so that there would be enough money to care for the Temple of our God; for we needed supplies of the special Bread of the Presence, as well as grain offerings and burnt offerings for the Sabbaths, the new moon feasts, and the annual feasts. 

2 Kings 4:23 And he said, Wherefore wilt thou go to him to day? it is neither new moon, nor Sabbath. And she said, It shall be well. 

Excerpted from the Quest Study Bible: The New Moon was both a religious and a civil festival. It was celebrated at the beginning of each month and is often mentioned in the Old Testament along with the Sabbath (for example, Isaiah 1:13). It was a day of celebration (Hosea 2:11), of rest (Amos 8:5), of increased offerings (Numbers 28:11-15), Ezekiel 45:17) and of worship (Isaiah 66:23; Ezek, 46: 1-7)

Book Excerpted: The Feasts of Israel (Seasons of the Messiah) by Bruce Scott published by Friends of Israel Gospel Ministry

Under the law of Moses, the new moon was marked by blowing two silver trumpets (Numbers 10:10) and sacrificing a burnt offering and a sin offering (Numbers 28: 11-15). Later, the beginning of each month took on additional characteristics. 

By the time of the Second Temple, the sighting of the new moon had become a festival in itself. Because only the Sanhedrin in Jerusalem had the authority to set the calendar, the sighting of the new moon was reported to them. Sumptuous meals were prepared in a large courtyard to encourage the people to serve as witnesses to these sightings. They gathered and waited their turn to be intervened by the religious leaders. Upon examination of two reliable witnesses, the Sanhedrin declared. 

Meanwhile, the high priest lifted his hands and blessed the people: "The Lord bless thee, and keep thee; the Lord make his face shine upon thee, and be gracious unto thee: the Lord lift up his countenance upon thee, and give thee peace. (Numbers 6: 24-26)

The Sanhedrin also gave three blessings. One over the holiday wine, one thanking God for revealing the Jewish calendar; and one expressing the desire that the Messiah come, along with his forerunner, Elijah the Prophet.

Numbers 28:11-15 "Also, on the first day of each month there shall be an extra burnt offering to the Lord of two young bulls, one ram, and seven male yearling lambs—all without defect. Accompany them with nine quarts of finely ground flour mixed with oil as a grain offering with each bull; and six quarts of finely ground flour mixed with oil as a grain offering for the ram; and for each lamb, three quarts of finely ground flour mixed with oil for a grain offering. This burnt offering shall be presented by fire and will please the Lord very much. Along with each sacrifice shall be a drink offering—six pints of wine with each bull, four pints for a ram, and three pints for a lamb. This, then, will be the burnt offering each month throughout the year. Living Bible

Hosea 2:11 I will also cause all her mirth to cease, her feast days, her new moons, and her Sabbaths, and all her solemn feasts.

Ezra 3:4. And they celebrated the Feast of Tabernacles as prescribed in the laws of Moses, sacrificing the burnt offerings specified for each day of the feast. They also offered the special sacrifices required for the Sabbaths, the new moon celebrations, and the other regular annual feasts of the Lord.

Isaiah 1:13 Bring no more vain offerings; incense is an abomination to me. New moon and Sabbath and the calling of assemblies— I cannot endure iniquity and solemn assembly.  Your new moons and your appointed feasts my soul hates; they have become a burden to me, I am weary of bearing them. ^RSV

Ezekiel 45:17 And it shall be the prince's part to give burnt offerings, and meat offerings, and drink offerings, in the feasts, and in the new moons, and in the Sabbaths, in all solemnities of the house of Israel: he shall prepare the sin offering, and the meat offering, and the burnt offering, and the peace offerings, to make reconciliation for the house of Israel. 

Amos 8:5 Saying, When will the new moon be gone, that we may sell corn? and the Sabbath, that we may set forth wheat...

Psalm 81: 3-7.  Sound the trumpet! Come to the joyous celebrations at full moon, new moon, and all the other holidays. 

^{1 Hillel, The Foundation for Jewish Campus Life website quote.}

2 Mesorah Publications' Stone Edition of the Chumash