Gemara

(Mishnah): One does not become a Ben Sorer u'Moreh if he ate with a group doing a Mitzvah, such as the meal to signify Ibur Chodesh (when a 30th day is added to the month).

Beitzah 15b (Rav Tachlifa - Beraisa): One's income is fixed, except for what he spends for Shabbos, Yom Tov and his children's Talmud Torah. If he decreases, they decrease for him. If he increases, they increase for him.

Berachos 49b (Rav Idi bar Avin): If one forgot to mention Rosh Chodesh in Birkas ha'Mazon, he does not go back.

(Rav Amram): Birkas ha'Mazon is not an obligation on Rosh Chodesh, i.e. one need not eat bread, therefore he need not repeat it.

Megilah 5a (Mishnah): The Megilah may be read early, but not later. The following may be postponed, but not advanced: Tish'ah b'Av...

Rishonim

Rif and Rosh (Megilah 3a and 1:7): Kri'as ha'Megilah may be advanced, but not postponed. The Seudah of Purim, of Rosh Chodesh, a Safek (eighth day for) Bris Milah... may be postponed, but not advanced.

Ran (DH Masnisin): Seudas Rosh Chodesh is when they agreed to make the previous month full (the Seudah publicized this).

Chidushei ha'Ran (standard edition, not in the Rav Kuk edition - Shabbos 24a DH v'Im): R. Yehonason says that on Rosh Chodesh one is obligated to eat bread and say Birkas ha'Mazon.

Hagahos Tur ha'Shalem (OC 419:1): Shirei Minchah points out that Berachos 49b refutes this! Acharonim say that R. Yehonason must mean like the Tur says, that it is a Mitzvah, but not a Chiyuv, to eat bread.

Roke'ach (228): One may not fast on Rosh Chodesh (Rosh Hashanah 19a). We find that Rosh Chodesh is called Mo'ed (Pesachim 77a). In Berachos, we say that eating is optional on Rosh Chodesh. This means that one need not be liable in a Zimun, for he may eat Peros if he wants. We do not wail (over a Mes) on Rosh Chodesh, for it is is equated to Mo'ed - "uv'Mo'adeichem uv'Roshei Chodsheichem." "I will return all its Simchah - its festival, (Rosh) Chodesh, Shabbos and all its Mo'adim. Sha'ul asked why David did not come to the Seudah, and Yehonason answered that David's family was having "Zevach Mishpachah." One does not become a Ben Sorer u'Moreh for eating at the meal of Ibur Chodesh. Pesikta of Sukos says that one's income is fixed on Rosh Hashanah, excluding what he spends for Seudos of Shabbos, Yom Tov, Rosh Chodesh, Chol ha'Mo'ed, and what one's children bring to their Rebbi's house.

Poskim

Shulchan Aruch (419): It is a Mitzvah to increase the Seudah of Rosh Chodesh.

Tur: The Yerushalmi says that Kri'as ha'Megilah and Shekalim are done early, but not late, but the Seudah of Purim, of Rosh Chodesh, a Safek Bris... are postponed, but are not done early. Since Seudas Rosh Chodesh is taught together with Seudas Purim, this shows that it is a Mitzvah. Also, Rosh Chodesh is equated to Mo'ed - "uv'Mo'adeichem uv'Roshei Chodsheichem." David said that his family was having "Zevach Mishpachah"; it was Rosh Chodesh. Pesikta (of Rav Kahana 28) says that one's income is fixed, excluding what he spends for Seudos of Shabbos, Yom Tov, Rosh Chodesh, Chol ha'Mo'ed, and what one's children bring to their Rebbi's house.

Bach (DH u'Mitzvah): The Tur must explain that we postpone Seudas Rosh Chodesh, i.e. when it is on Shabbos, the Seudah is on Sunday, just like Seudas Purim is postponed until Sunday when Purim is on Shabbos. Why did the Tur need to learn from the Hekesh? If there is no Mitzvah, what is postponed?! One could have said that it is only a custom.

Beis Yosef (DH u'Mitzvah): According to the Ran, there is no proof (Seudas Ibur Chodesh is a Mitzvah; it is postponed)! The Hekesh to Mo'ed is a mere Semach (it is not a Drashah teaching about Seudas Rosh Chodesh). We need not say that "Zevach Mishpachah" was due to Seudas Rosh Chodesh. Perhaps it was a Seudah of comradery, and it happened to be on Rosh Chodesh! Also, perhaps David merely told Yehonason to say this, but really, there was no Seudah! However, since David suggested telling this to Sha'ul, this is some support they had such a custom. The Roke'ach says that it is a Mitzvah to eat on Rosh Chodesh, and brought several proofs.

Bach (DH u'Veis): The Ran has no source to say that it refers to Seudas Ibur Chodesh. Since there is a Hekesh to Mo'ed, presumably it discusses any Seudah of Rosh Chodesh. We find that one's allotted income does not include Seudos Rosh Chodesh. The Tur (Siman 420) brings the Hekesh to show that Rosh Chodesh is a day of Simchah, and we say Hallel - "Zeh ha'Yom... Nagilah..." This is the Halachah, that one should increase his Seudah on Rosh Chodesh, and on Sunday when Rosh Chodesh is on Shabbos; however, people do not do so.

Magen Avraham (1): The Rif and Rosh say that when Rosh Chodesh is on Shabbos, the Seudah is postponed, for it would not be recognized that it is for Rosh Chodesh. The Bach (688) says that we make Seudas Purim on Shabbos. I say that on Rosh Chodesh, Seudah Shlishis should be extended into the night, to fulfill Seudas Rosh Chodesh.

Sha'ar ha'Tziyon (4): Nowadays, people often extend Seudah Shlishis into the night, so it is not evdent that it is for Rosh Chodesh.

Sha'ar ha'Tziyon (5) and Kaf ha'Chayim (3): The Ya'avetz says to increase the fourth Seudah (Melaveh Malkah) more than usual to fulfill Seudas Rosh Chodesh.

Darchei Moshe (1): Or Zaru'a (453) says that one may not fast on Rosh Chodesh, but he need not eat bread. It is praiseworthy to spend money on Seudas Rosh Chodesh, and eat and drink well.

Mishnah Berurah (2): It suffices to increase the day Seudah. One need not increase at night. Meticulous people have one food more than they usually do on a weekday (or Shabbos) when Rosh Chodesh is on a weekday (or Shabbos), so Kavod Rosh Chodesh will be evident.

Kaf ha'Chayim (4): An Oni should buy at least a fruit for Rosh Chodesh. Men and women are equally obligated in Seudas Rosh Chodesh.

Kaf ha'Chayim (5): The Seudah should be eaten honorably, on a table, like Seudas Shabbos.

Kaf ha'Chayim (6): One should change clothes for Rosh Chodesh. Yir'ei Shamayim wear special clothes to honor Rosh Chodesh. It is good to have four levels of clothing, for weekdays, Shabbos, Yom Tov, and (the fourth level includes) Rosh Chodesh, Chol ha'Mo'ed, Purim and Chanukah.

Beis Yosef (DH v'Isa): 'What one's children bring to their Rebbi's house' refers to his wages for teaching them, like it says in Beitzah. Our Gemara does not mention expenditures for Rosh Chodesh. We can say that it is included in Yom Tov.

Rebuttal (Prishah 1): 'What one's children bring to their Rebbi's house' does not connote wages. Rather, it is gifts and food. Everything that the Pesikta discusses refers to food.

Kaf ha'Chayim (2): Even though one is Yotzei through Peros, Yir'ei Shamayim fulfill all opinions (and eat bread). One should bless on two loaves to honor Rosh Chodesh more than a weekday. One who honors Mitzvos will be rewarded doubly.

Kaf ha'Chayim (7,8): It is good to light candles on Rosh Chodesh, but not as nice as on Shabbos and Yom Tov. One should give Tzedakah on Rosh Chodesh, just like on Erev Shabbos and Erev Yom Tov.

Aruch ha'Shulchan (2): Why do we have no custom for Seudas Rosh Chodesh, especially when it is on Shabbos? Perhaps it is because the Bavli did not say that it is not included in one's allotted income. Also, we say that Seudas Ibur Chodesh is a Mitzvah, which implies that Stam Seudas Rosh Chodesh is not.

Aruch l'Ner (70b DH Seudas): We can say that the Mitzvah of Seudas Rosh Chodesh is only during the day, like Seudas Purim, and Seudas Ibur Chodesh was at night. The Rambam (Hilchos Kidush ha'Chodesh 3:7) says that Seudas Ibur Chodesh is during the day, and 'this is Seudas Ibur Chodesh mentioned everywhere.' He must mean (like the Ran) that this is Seudas Rosh Chodesh that the Yerushalmi mentions, for it is not mentioned anywhere else! The Magen Avraham holds that it is a Mitzvah, and it is even at night. He must say that the Mishnah exempts even for Ibur Chodesh even if he already ate at night (so it is not a Mitzvah due to Rosh Chodesh)