“In the beginnings of your months.” This could also be translated “at your new moons” (cp. JPS; TNK). Each month was dedicated to God with special sacrifices as we see here in Numbers 28:11-15, and with the blowing of trumpets (Num. 10:10; Ps. 81:3). Although God did not specifically designate it in the Law of Moses as a national feast or day of rest, in time that came to be the accepted practice.
Numbers 28 and 29 contain a list of the occasions when sacrifices and offerings were offered to God, and these included the daily sacrifices and offerings (Num. 28:1-8), the Sabbath offerings (Num. 28:9-10), the new moon offerings (Num. 28:11-15); the Passover and Feast of Unleavened Bread (Num. 28:16-25); the Feast of Weeks also called the Day of Pentecost (Num. 28:26-31); the Feast of Trumpets (Num. 29:1-6); the Day of Atonement (Num. 29:7-11); and the Feast of Tabernacles (Num. 29:12-38).
The new moon, with its sacrifices and then later with its additional trappings of being a feast day and a day of rest, shows up a number of times in Scripture (cp. 1 Sam. 20:5, 18, 24; 2 Kings 4:23; 1 Chron. 23:31; 2 Chron. 2:4; 8:13; 31:3; Ezra 3:5; Neh. 10:33; Isa. 1:13; 66:23; Ezek. 45:17; 46:1; Hosea 2:11; Amos 8:5; Colossians 2:16).