(יא) וּבְרָאשֵׁי חָדְשֵׁיכֶם תַּקְרִיבוּ עֹלָה לַיהוָה פָּרִים בְּנֵי בָקָר שְׁנַיִם וְאַיִל אֶחָד כְּבָשִׂים בְּנֵי שָׁנָה שִׁבְעָה תְּמִימִם.
We are thus introduced to Rosh Chodesh. There is no Issur Melakha and no special mitzvah for the day, with its only defining factor being the Korban Musaf. Let's see how this day was observed/preserved by the ancients.
Scenario 1: Tekiat Shofar
(ד) תִּקְעוּ בַחֹדֶשׁ שׁוֹפָר בַּכֵּסֶה לְיוֹם חַגֵּנוּ.
(4) Blow the horn at the new moon, At the full moon for our feast-day.
(ד) אע"פ שתקיעת שופר בראש השנה גזירת הכתוב, רמז יש בו, כלומר עורו ישינים משנתכם ונרדמים הקיצו מתרדמתכם וחפשו במעשיכם וחזרו בתשובה וזכרו בוראכם...
(4) Even though the blowing of the shofar on Rosh HaShanah is a Biblical decree, it hints at something, i.e., “Wake up, sleepers, from your sleep! And slumberers, arise from your slumber! Search your ways and return in teshuvah and remember your Creator!...
מוסף לראש חודש
ראשי חדשים לעמך נתת, זמן כפרה לכל תולדותם.
Musaf for Rosh Chodesh
Beginnings of Months you gave to your people, a time of atonement for all their deeds.
Although most of us nowadays do not use Rosh Chodesh in this fashion, in the days of David it was apparently the Minhag Yisrael to pray for atonement at the beginning of the month. This tradition also surfaced much later on, in the Kabbalistic city of Tzefat; see the following link.
http://en.wikipedia.org/wiki/Yom_Kippur_Katan
Scenario 2: Yehonatan and David
(18) And Jonathan said unto him: ‘To-morrow is the new moon; and thou wilt be missed, thy seat will be empty.
Question: How could Yehonatan know when the new moon was? They didn't have a fixed calendar, so Rosh Chodesh was declared by the Sanhedrin
(ז) ראש בית דין אומר מקדש, וכל העם עונין אחריו מקדש מקדש.בין שנראה בזמנו בית שלא נראה בזמנו, מקדשין אותו.רבי אלעזר ברבי צדוק אומר, אם לא נראה בזמנו, אין מקדשין אותו, שכבר קדשוהו שמים.
(7) The chief of the tribunal [בית דין] then said, the feast of the new moon is mekoodash [consecrated]; and all the people said after him, "Mekoodash, mekoodash." Whether the new moon had been seen at the proper time or not it was consecrated. R. Eleazar, son of Zadok, said, "If it had not been seen at its proper time, it was not consecrated, because it had already been consecrated in heaven."
What the underlined text is saying is that if Rosh Chodesh is not on the thirtieth, then it is already plainly obvious that the next day is the new moon. That knocks out our top question.
(כד) וַיִּסָּתֵר דָּוִד בַּשָּׂדֶה וַיְהִי הַחֹדֶשׁ וַיֵּשֶׁב הַמֶּלֶךְ על [אֶל] הַלֶּחֶם לֶאֱכוֹל.
(כז) וַיְהִי מִמָּחֳרַת הַחֹדֶשׁ הַשֵּׁנִי וַיִּפָּקֵד מְקוֹם דָּוִד וַיֹּאמֶר שָׁאוּל אֶל יְהוֹנָתָן בְּנוֹ מַדּוּעַ לֹא בָא בֶן יִשַׁי גַּם תְּמוֹל גַּם הַיּוֹם אֶל הַלָּחֶם.
(27) And it came to pass on the morrow after the new moon, which was the second day, that David’s place was empty; and Saul said unto Jonathan his son: ‘Wherefore cometh not the son of Jesse to the meal, neither yesterday, nor to-day?’
The bolded text is there to point out something strange. It seems that Shaul is observing two days of Rosh Chodesh, having a special holiday meal for two days in a row.
You could think that the second day was Shabbat. However:
Weapons are Muktzah on Shabbat, so clearly this second day is a different festival.
Next option: Maybe it was Rosh Hashanah, which is celebrated for two days even in Israel.
However, Rosh Hashanah also has the Issur Melakha. So, there seems to be no answer.
(ד) בראשונה היו מקבלין עדות החדש כל היום, פעם אחת נשתהו העדים מלבוא, ונתקלקלו הלוים בשיר, התקינו שלא יהו מקבלין אלא עד המנחה.ואם באו עדים מן המנחה ולמעלה, נוהגין אותו היום קדש ולמחר קדש...
(4) Formerly evidence as to the appearance of the new moon was received the whole [of the thirtieth] day; but as it once happened that the witnesses delayed coming, which interrupted the Song of the Levites [at the evening sacrifice], it was ordained that witnesses should be admitted till the time of Minchah only, and if any came after that time, that and the following day were kept as holy...
So, they actually did keep two-day Rosh Chodesh! However, here's the new question: We said that David and Yehonatan talked on the thirtieth. That would mean that we have a day of Rosh Chodesh that couldn't be so!
(א) ... וכגין מועדי ה' אשר תקראו אתם, שנכתב חסר בשלשה מקומות, ללמד, אתם אפילו שוגגים, אתם אפילו מזידים, אתם אפילו מוטעים :
... Just like, "These are the festivals of G-d, which you shall proclaim [in their appointed times]," for it was written missing [in their appointed times] three times, to teach: Even if you make a mistake, even if you purposely [get it wrong], even if you make a mistake [these are still my holidays].
Although we may have messed up, the holidays are still holy days.
Hope you liked it!