The Hebrew word rosh means "head". The word chodesh means "month". Thus the name translates to "head of the month".
Rosh Chodesh, the first day of each Hebrew month, is declared a holiday in Numbers 10:10 as part of the Sinai/Moav covenant. It was commanded to the Israelites as a reflection of heavenly worship (Hebrews 8:5, 9:24), so we can trust it is celebrated in heaven now.
Nevertheless, the command to celebrate Rosh Chodesh is not said to be "for all time" or "for all generations", so it is optional for us today, while there is no Temple.
Paul taught that this holiday was optional for Gentile believers even when the Temple was still in use: Gentile believers are free not to celebrate according to Colossians 2:16-17.
Let no man therefore judge you in eating, or in drinking, or with respect to a feast day or Rosh Chodesh or Shabbat, which are a shadow of the things to come; but the body is Messiah's.
According to Isaiah 66:23, in the millennial reign all people will observe Rosh Chodesh.
It shall happen, that from one Rosh Chodesh to another, and from one Shabbat to another, shall all flesh come to worship before me, says Adonai.
Rosh Chodesh was celebrated at the place dedicated to God where people gathered to worship. Originally this was the Tabernacle, and then the Temple. Today this is the sanctuary.
Behold, I am about to build a house to the name of Adonai my God, to dedicate it to him, and to burn before him incense of sweet spices, and for the continual show bread, and for the olah offerings morning and evening, on Shabbat, and on Rosh Chodesh, and on the moadim of Adonai our God. This is an ordinance forever to Israel.
-Second Chronicles 2:4
We need not do a full Shabbat service, but we should shachah—worship and prostrate ourselves before God—in the same way we do at Shabbat.
The people of the land shall worship (shachah) at the door of that gate before Adonai on Shabbat and on Rosh Chodesh.
Although Rosh Chodesh is not one of the moadim and not listed in Leviticus 23, it is similar to Shabbat at least in worship and also avoiding economic activity.
[You are] saying, 'When will Rosh Chodesh be gone, that we may sell grain? And Shabbat, that we may market...'
Also, like Shabbat, Rosh Chodesh was a time of delight.
I will also cause all her delights to cease: her pilgrimage festivals, her Rosh Chodesh, her Shabbat, and all her appointed times.
In Numbers 28:11-15 the people are commanded to bring certain offerings on Rosh Chodesh:
In Ezekiel 46:1-8, the "Prince" (Messiah) is commanded to bring only some of these Rosh Chodesh offerings in the Messianic age:
Note that the Prince is doing the work of the people: in the Messianic age he will bring sacrifices for them. The Messiah's offerings in the Messianic age will be an earthly manifestation of his ongoing work advocating for us before the Father (First John 2:1). This was stated earlier in the previous chapter of Ezekiel:
It shall be the Prince's part to give the olah, minchah, and drink offerings at each pilgrimage festival, and on Rosh Chodesh, and on Shabbat, in all the appointed times of the house of Israel: he shall make the chattat, minchah, olah, and sh'lamim offerings, to provide kofer (covering-atonement) for the house of Israel.
Also notice that Ezekiel 46:2 says that the priests will prepare the offering the Prince brings. Even now, we who are believers are God's current priesthood (First Peter 2:5) and must do some preparatory work (humble repentance) before our Prince can intercede for us.
We can ask why the Prince brings one less bull and lamb in the Messianic age? And why does he not need to offer the goat chattat or the wine drink offering? Actually, these four items are not missing because our Prince himself represents each of these things: his sacrifice was like a bull as olah since it was costly and connects us with God; it was like a goat chattat since it provided forgiveness of sins; it was like pesach lamb which causes death to pass over us; it was like spilled out wine. The omission of these four specific items reminds us of all the different things the Messiah's death provides.
In other words, in the Messianic age the offering at Rosh Chodesh serves as a reminder of the Messiah's sacrificial death, officiated by the Messiah himself after his priests prepare it. Thus Rosh Chodesh seems a fitting time to celebrate the Lord's Supper.
(I share this cautiously, aware that the same thinking does not explain why the Prince, in Ezekiel 46:4-5, is instructed to offer on Shabbat four lambs and one ram more than the people were commanded to bring on Shabbat in Numbers 28:9-10. Nevertheless, since First Corinthians 11:26 allows us to celebrate the Lord's Supper as often as we wish, my interpretation does no harm even if incorrect.)
(The Rosh Chodesh offering is also mentioned in Numbers 29:6, Ezra 3:5, Nehemiah 10:32-33, First Chronicles 23:31, and Second Chronicles 8:13 and 31:3.)
In the Talmud (Bereishit Rabbah 15:24) some ancient rabbis linked together Jeremiah 33:25, Leviticus 20:8, and Exodus 12:2 to create a teaching that it is Israel who makes the new month holy, and by regularly helping God make part of creation holy the nation helps God to continually renew the world. This seems to be a likely case of borrowing from pagan Babylonian traditions, which Messianic believers need not apply. But we can see a link: in the Lord's Supper we are also participating in Yeshua's renewing of the world.
The silver trumpets of Numbers 10:2 were blown (with rejoicing) over the sacrifices on all holidays, including both the moadim and Rosh Chodesh.
Also in the day of your rejoicing, and in your moadim, and in Rosh Chodesh, you shall blow the trumpets over your olah and sh'lamim offerings. They shall be to you for a remembrance before your God; I am Adonai your God.
Since the Lord's Supper is our current sh'lamim (fellowship meal with God that allows God to dwell within us and spread his holiness) perhaps we should blow a trumpet over it. But more likely trying to do this would be inappropriately mimicking Temple worship. We do not have the silver trumpets, and are not literally bringing those types of offerings.
The court of King Saul ate together at least at Rosh Chodesh and the day after (see First Samuel 20:5-34). This may or may not mean it was a God-ordained thing to do.
In Second Kings 4:23 we read that during that time people would seek out a prophet on Rosh Chodesh or Shabbat. This may or may not mean it was the God-ordained thing to do. Perhaps we should also have a time of seeking God's will on Rosh Chodesh, in the same ways we do on Shabbat.
The image we are given of the new heaven includes the tree of life:
On this side of the river and on that was the tree of life, bearing twelve kinds of fruits, yielding its fruit every month. The leaves of the tree were for the healing of the nations.
Perhaps we should be seeking a renewed nourishment of Eternal Life at Rosh Chodesh, or praying for the nations.
Twice in the Talmud (Pirkei deRabbi Eliezer 45, Megillah 22b) the rabbis teach that women should be exempt from household work in Rosh Chodesh in reward for refusing to participate in the golden calf episode. This may be the most widespread way rabbinical Judaism still observes Rosh Chodesh.
Psalm 81:3 is not about Rosh Chodesh. Both Pesach and Sukkot happen on the 15th day of the month, on the full moon. These two moadim and Shavuot are the three pilgramidge festivals, called a chag in Hebrew. Psalm 81:3 says "Sound in the month the shofar in the full moon to the day of our chag." This is about sounding a shofar at either Pesach or Sukkot, not at Rosh Chodesh.
Rosh Chodesh is not the first commandment given to the Israelites as a nation. In Exodus 3:12 God tells Moshe to bring the new nation to Mount Sinai. The high point of the "Song of the Sea" in Exodus 15:17-18 is recalling this first commandment to be "planted" at Mount Sinai. Exodus 12:2 does mention a month, but this verse contains no command to celebrate the new moon in any ways.
The start of the new month was communicated by beacons from hilltops: long poles of cedar, topped with a torch of reeds, olive wood, and flax fluff (Rosh HaShana 22b). Bonfires are only associated with Lag B'Omer.