In honor of Rosh Chodesh we call up four people to the Torah. The reading begins with the daily burnt offering and ends with the special Rosh Chodesh sacrifices (Bamidbar, 28:1-15). This hints to the fact that the special sanctity of Rosh Chodesh, which incorporates renewal, atonement, and repentance, stemmed from the fixed and continual sanctity that manifests itself in the daily burnt offering, which was sacrificed every morning and afternoon.
After the Torah reading and the recitation of Ashrei and U’Va LeTzion (and according to some Sefardim, Beit Ya’akov and the daily Psalm, as well), we say the Silent Prayer of Musaf. This prayer is comprised of the usual three opening and closing blessings, with one middle blessing about Rosh Chodesh, which concludes, “Blessed are You, HaShem, Who sanctifies Israel and the beginnings of the months.”
Our Sages instituted the Musaf service in place of the additional musaf sacrifices that were offered in the Beit HaMikdash on Rosh Chodesh. The time frame for saying this prayer also corresponds to that of the additional sacrifices. Thus, one must pray Musaf by the end of the seventh hour of the day, calculating each “hour” as one-twelfth of daylight. One who procrastinates and fails to pray by this time is considered a sinner. Nevertheless, he should pray afterwards, because, b’dieved (ex post facto), one may offer the additional sacrifices all day long (Shulchan Aruch 286:1).
The custom is to remove one’s tefillin before the Musaf prayer. On Yom Tov (major Jewish holidays) we do not wear tefillin at all. Since the day itself is a “sign” (ot) between God and Israel, we do not need the additional “sign” of tefillin. Similarly, the Musaf prayer on Rosh Chodesh is considered a “sign,” and we do not need the additional “sign” of tefillin(Shulchan Aruch 423:4, Mishna Berura 10). The custom is to remove them after the Kaddish that precedes Musaf. It is proper to start praying only after the tefillin are completely wrapped in their straps and stored away in their bags. Otherwise, they will lie there throughout the Musaf service, without receiving the respect demanded by their great holiness.
Another prevalent custom is to say the Psalm beginning with Borchi Nafshi(Tehillim 104) during Shacharit, because it contains the verse, “He made the moon for appointed festivals.” Some believe that the Levites recited this Psalm in the Beit HaMikdash on Rosh Chodesh(Aruch HaShulchan 423:5).
 The Beit Yosef (25:13) gives a different reason, stating that it is inappropriate to be without the crown of tefillin when saying the stanza beginning with Keter (“crown”) in the Kedushah of Musaf. However, even Ashkenazim, who omit Keter, are accustomed to removing their tefillin before Musaf. Therefore, I cited the Levush’s explanation that Musaf is considered a “sign,” similar to Yom Tov, on which we do not wear tefillin (Shulchan Aruch, Orach Chaim 31:1). See Sefer Rosh Chodesh 8:1. The prevalent custom is to remove the tefillin after Kaddish (according to the Ba’al HaTanya’s siddur, they are removed before Kaddish). See Sefer Rosh Chodesh 8:4-6.
This raises a question regarding the common practice to don tefillin at the circumcision of one’s son, in order to be surrounded by two signs, the sign of the brit and tefillin. Do we not learn from the law of tefillin on Shabbat and Yom Tov that one should not display two signs together, for each one implies that the other is insufficient, thus degrading its significance (Shulchan Aruch 31:1)? The Eliyah Rabbah (29) answers in the name of the Roke’ach that the sign of circumcision is not in commemoration of the Exodus from Egypt; therefore, it is good to combine it with another sign that is. Tefillin, Shabbat, and Yom Tov, however, are in commemoration of the Exodus and should not be joined together.
 For the order of the Torah reading, see Tur, Beit Yosef, Shulchan Aruch, and Mishna Berura 423:2; Yalkut Yosef 423:4; Sefer Rosh Chodesh 7:9. If one said Musaf before Shacharit, he has discharged his obligation, but ideally one must pray Shacharit first, just as one must sacrifice the daily burnt offering before all other offerings (Rama 286:1).
For more on Borchi Nafshi, see Sefer Rosh Chodesh 7:5. The following authorities hold that Borchi Nafshi was the daily song for Rosh Chodesh in the Temple: The Vilna Gaon (Ma’aseh Rav 157), B’nei Yissachar (Rosh Chodesh 3:1), and Aruch HaShulchan (423:5, 424:3). Ashkenazim and Chassidim (Nusach Sefard) say Borchi Nafshi after the regular daily song; the Chassidim recite both songs after Hallel; and the Ashkenazim after Musaf (Sefer Rosh Chodesh 7, n. 5). Some Sefardim have the custom of saying the daily song after U’Va LeTzion, before the Kaddish that precedes Musaf, and Borchi Nafshi after Musaf. Others say Borchi Nafshi after Musaf and omit the daily song altogether.
The Ashkenazim customarily return the Torah scroll to the Ark immediately after it is read (Mishna Berura 423:5). The Shulchan Aruch (423:3) writes that it should be returned after U’Va LeTzion, and Sefardim and Chassidim follow this practice. (However, the author of Kaf HaChaim [135:2 and 423:11] agrees with the Ashkenazi custom, and Rav Ovadyah Yosef follows it for Kabbalistic reasons, returning the Torah scroll, on Rosh Chodesh and Mondays and Thursdays, immediately after the reading [Yalkut Yosef 423:6 with notes].)