One must stand while reciting the Hallel, because Hallel is a testimony to God’s glory, and witnesses must stand while testifying. After the fact, if one said Hallel sitting or lying down, he has nonetheless fulfilled his obligation. Someone who is ill and cannot stand may say it sitting or lying down from the outset (Shulchan Aruch 422:7, Mishna Berura 28).
One should not interrupt the recitation of Hallel, even by just remaining silent. In the case of a pressing need, however, like preventing an insult, one may interrupt. One may also interrupt Hallel in order to recite “holy responses” [like Kedushah, Kaddish, and Barachu]. One should say Hallel in order, from beginning to end. One who said it out of order has not fulfilled his obligation and must go back to the place where he erred and read it in its proper sequence (Shulchan Aruch 422:4-6). It is proper to read Hallel slowly and pleasantly, and many congregations have a custom to sing portions of the verses.
Our Sages ordained that Hallel be said immediately after Shacharit prayers After all, we mention the uniqueness of Rosh Chodesh in the Silent Prayer, by saying Ya’aleh VeYavo. Therefore, it is appropriate to [immediately] continue praising God and thanking Him for sanctifying the Jewish people and the New Moons. Under extenuating circumstances, one may say it later in the day, because, according to the letter of the law, the entire day is suitable for the reading of Hallel(Megillah 20b).
There are various customs surrounding the Hallel’s recital, regarding which verses are repeated twice, and which verses are said responsively, etc. All the customs are proper, and every community should continue following its custom (Sukkah 38a-39a; Shulchan Aruch 422:3).
Customarily, the cantor reads four verses aloud (Tehillim, 118:1-4): 1) הודו לה’ כי טוב כי לעולם חסדו – Give thanks to the Lord for He is good, for His kindness endures forever. 2) יאמר נא ישראל כי לעולם חסדו – Let Israel say now, “For His kindness endures forever.” 3) יאמרו נא בית אהרן כי לעולם חסדו – Let the House of Aharon say now, “For His kindness endures forever.” 4) יאמרו נא יראי ה’ כי לעולם חסדו – Let those who fear the Lord say now, “For His kindness endures forever.” According to Ashkenazi custom, the congregation responds,הודו לה’ כי טוב כי לעולם חסדו to each of these verses. According to the Sefardic custom, however, the congregation repeats each verse after the cantor.
Regarding the duplication of verses, it has become the accepted custom, in the last few generations, to repeat all the verses from אודך until the end of Hallel(Tehillim, 118:21-29). The reason we say these verses twice is that the beginning of the chapter repeats every idea twice, but from אודך on, the verses cease repeating themselves. We, however, continue the pattern of the Psalm and double up the rest of the verses. Furthermore, King David, his father Yishai, and his brothers composed these verses, as the Talmud relates (Pesachim 119a). Thus, because of their importance, [Chazal] wanted us to say them twice.
We read the verse, אנא ה’ הושיעה נא, אנא ה’ הצליחה נא – “Please, O Lord, save us; Please, O Lord, make us successful”(Tehillim, 118:25), in a special way, saying the first part twice, and then the second part twice.
Rav Amram Gaon mentions both customs in his siddur, the first one as the Sefardic custom, and the second as the Ashkenazi practice. Tosafot and the Ran (Sukkah 38b) also cite the Ashkenazi custom [as described above], as do the Tur and Beit Yosef 422:3. The congregation fulfills their obligation to say [the other three verses] by hearing the cantor chant them out loud. Therefore, they can simply answer הודו לה’ כי טוב כי לעולם חסדו. Many Acharonim write that since there is reason for concern that some congregants may not hear the cantor properly, it is best for the congregation to say the verses along with the cantor, finish shortly before him, and then answer הודו לה’ כי טוב כי לעולם חסדו. This is the opinion of Magen Avraham 422:8, Eliyah Rabbah 13, Machatzit HaShekel and Mishna Berura 20. See also Sefer Rosh Chodesh 6:15.
 According to Sefardic tradition, the cantor, followed by the congregation, says the first part twice and then the second part twice. The Ashkenazi custom is that the cantor and the congregation say each verse responsively. This raises a question. The Talmud states in Megillah 22a that one is forbidden to interrupt a verse in the middle, except for the purpose of teaching young children. Tosafot (Sukkah 38b) answer that two people composed this verse – David and his brother. The Kolbo suggests that one is forbidden to split a verse in two only if it is from the Torah (Magen Avraham 422:8). The Maharsham explains in Da’at Torah that one is permitted to divide a verse if it is being used for prayer. See Sefer Rosh Chodesh 6:18, n. 37.
Ashkenazim say the four verses beginning with the words אודך, אבן מאסו, מאת ה’, זה היום twice. The custom of many Jews from North Africa is that the cantor says each verse once, after which the congregation responds likewise, and since “hearing is like answering,” it is considered as if they said each verse twice.