Rosh Chodesh and New Beginnings

Rosh Chodesh means "head of the month." The month begins with the appearance of the new moon (Rosh Chodesh). The ancient Rabbis understood Exodus 12:1-2 as requiring that the new month be determined by eyewitness testimony: just as God "pointed out" the appearance of the new moon to Moses and Aaron, so must witnesses declare before the judges of the Sanhedrin, the Great Court in Jerusalem, who stood in place of Moses and Aaron, that they had seen that form in the heavens.
While eyewitness testimony was a legal and ritual requirement, the Sages knew how to determine the calendar by means of astronomical and mathematical calculations. They also knew that the duration of the moon's cycle is twenty-nine and one-half days and that the new moon would be seen on either the thirtieth or thirty-first day following the previous Rosh Chodesh. Following the destruction of the Temple and the disappearance of the Great Court, the eyewitness procedure was abandoned, and these formulae became the exclusive means by which the Jewish calendar was set.
In biblical times, Rosh Chodesh, the first of every month, was a festival marked by joyous sacrifice, feasting, and trumpets.
from ReformJudaism.org

Text #1
The Distinction Between Rosh Chodesh and Shabbat and Chagim--
Rosh Chodesh: A Lesson in Renewal by Rabbi Eliyahu Safran
Wherein lies the mystery and mystique of Rosh Chodesh?
Rav Soloveitchik teaches that while Shabbat and Yom Tov receive their sanctity and significance from specific historical-reli­gious events and eras, such as Creation, Revelation or the Exodus from Egypt, the significance of Rosh Chodesh emanates directly from the Jewish embrace of Renewal. The Jew identifies personally and dynamically with the moon; with revival from an almost nonex­istent state, with illumination returning from a state of darkness. The Talmud alludes to this when it teaches: sanctify the moon, and I will send you a sign – David Melech Yisrael chai vekayam! David, who was defeated and humiliated, lives! He will rise! He will reign with power and pride. Moshiach will be ben David."

Text #2
The moon, when she was first created, was a glistening jewel. She did not merely reflect light, but rather transformed it and brought out its inner beauty, much as a precious stone glistens with a secret, hidden light all its own. In her own way, the moon was greater than the sun—for the sun only shines from its surface, whereas the moon shone from its inner essence. The sun holds the light that extends outward, whereas the moon holds the light of being.
And so will be, once again, and much more so, in the time to come, once we have transformed the world with the Torah and its mitzvahs. (R. Isaac of Homil, Shnei Me’orot)

Text #4
Throughout history, the Jewish people have had a connection to the moon. Other ancient cultures revered the sun. A smaller nation, we saw ourselves in the small moon. Just like God, the moon would always return providing us with the safety of its protective light. (reformjudaism.org)

... sometimes, beginnings don’t need major deliberation or fanfare. The start of a journey can be as easy as turning a page or reversing the hourglass. It’s our own mental and emotional blocks that often inhibit beginnings to occur. But if we model the reading of Torah, the restarts we want to see in our lifetime can be seamless. One fluid motion, briefly acknowledging an ending without getting stuck, rooted, unable to step forward.
We have permission to redo, jump back in, try another time, begin again. Whether we want to reimagine a relationship, go back to school, try a new profession, take on a hobby, or start a journey, the Jewish calendar focuses on movement. A fleeting pause between Moses’ death and the art of creation. Time is given to live out the next chapter. Verse by verse, we learn, we grow, we live.
Don’t take too long to begin. Again. There is no better time than right now. - Rabbi Nicole Guzik