Numbers 10:10: Also in the day of your gladness, and in your solemn days, and in the beginnings of your months, ye shall blow with the trumpets over your burnt offerings, and over the sacrifices of your peace offerings; that they may be to you for a memorial before your God: I am the LORD your God.

Numbers 28:11-15: And in the beginnings of your months ye shall offer a burnt offering unto the LORD; two young bullocks, and one ram, seven lambs of the first year without spot; And three tenth deals of flour for a meat offering, mingled with oil, for one bullock; and two tenth deals of flour for a meat offering, mingled with oil, for one ram; And a several tenth deal of flour mingled with oil for a meat offering unto one lamb; for a burnt offering of a sweet savour, a sacrifice made by fire unto the LORD. And their drink offerings shall be half an hin of wine unto a bullock, and the third part of an hin unto a ram, and a fourth part of an hin unto a lamb: this is the burnt offering of every month throughout the months of the year. And one kid of the goats for a sin offering unto the LORD shall be offered, beside the continual burnt offering, and his drink offering.

Being a lunar calendar, the new moon marked the beginning of each new month.

From the LDS BIBLE DICTIONARY - Feasts:
The law also directed that at the New Moons special sacrifices should be offered (Num. 10: 10; Lev. 23: 24-25; 1 Sam. 20: 5-6, 29; 2 Kgs. 4: 23; Amos 8: 5). As the days of the celebration of all the great Feasts of the Jews were reckoned by the moon, the exact time of the appearance of the new moons was a matter of importance. Watchers were placed on the heights around Jerusalem to bring the news of its appearance with all speed to the Sanhedrin, who proclaimed it as soon as satisfactory evidence was given. Watchfires on the hilltops told the news to distant cities. It is said the Samaritans, to cause confusion, lighted fires at wrong times. See also Jubilee, Year of; Sabbatical year.

From PELOUBET'S BIBLE DICTIONARY - New Moon:
The first day of the lunar month was observed as a holy day. In addition to the daily sacrifices there were offered to young bullocks, a ram and seven lambs of the first year as a burnt offering, with the proper meat offerings and drink offerings, and a kid as a sin offering. As on the Sabbath, trade and handicraft work was stopped, and the temple was opened for public worship. Trumpets were blown at the offering of the special sacrifices for the say, as on the solemn festivals.