The Torah reading for every day of Chanukka are the offerings of the princes in the Naso portion [Bamidbar, 7.1]. The reason given in the Pesikta is that the work on the Mishkan (Tabernacle) was completed on the twenty fifth day of the month of Kislev, as I have previously mentioned in chapter 670.
The reading begins at the words "ויהי ביום כלות משה" - "It was on the day Moshe finished...", which is the beginning of the account. There are locales where the custom was to begin at the priestly blessings [6.22] beacuse the miracle of Chanukkua occurred through the agency of Kohanim [members of the priestly class], and the Tur describes it as a 'beautiful custom'. Even so, we do not accustom ourselves so since these verses are not part of the subject matter of the princes' offerings.
The order of the readings is as follows:
One the first day one begins at "ויהי ביום כלות משה" - "And it was on the day that Moshe completed...". This sections has many verses and is apportioned between the Kohen and Levi. The third reading is the portion of the price of Judah's offering. Some say instead that the Kohen is given the entire portion of "ויהי ביום כלות משה" - "And it was on the day that Moshe completed...", while the Levi and Yisrael divide the verses regarding the prince of Judah's offering. Their reason is that the purpose of this reading is to recount the offerings of the tribal princes, and therefore the majority of the readings should be split among a prince's offering. This is our custom.
One the second day the Kohen reads the from the words "ביום השני" - "On the second day" until "פר אחד בן בקר" - "a cow from among the cattle". The Levi then reads from "פר אחד" - "one cow" until "ביום השלישי" - "On the third day". A Yisrael then repeats the portion of "ביום השני" - "On the second day". Others say that the Yisrael reads the portion "ביום השלישי" - "One the third day", since it preferable to read the following day's portion rather than repeat the previous readings, if possible. A similar approach is followed on every one of the days.
On the eighth day the Kohen and Levi are apportioned the section "ביום השמיני" - "On the eighth day", since that is the day's obligatory offering. The Yisrael then reads from "ביום התשיעי" - "On the ninth day" until the conclusion of the princes' offerings, as well as the section "זאת חנוכת המזבח" - "This was the inauguration of the Altar", and finally the section of "בהעלתך" - "When you kindle the lamps...[of the Menorah in the Temple]" until "כן עשה את המנורה" - "so did he construct the Menorah", as this is all of one subject and bears reading until its completion.
When Chanukkah falls out on Shabbos two Torah scrolls are used for the readings. From one the Torah portion of the week is read, which generally will be the portion of Miketz. From the other the Chanukkah portion for that day is read as the Maftir. The Haftorah read is the vision of the candelabra in Zecharia [Zechariah 2.14-4.7], starting from "רני ושמחי" - "Sing and rejoice". If the first day of Chanukkah falls out on Shabbos, so that Chanukkah will fall on Shabbos twice, the Haftorah read on the second Shabbos is that of the Menorah of King Solomon, starting with "ויעש חירום" - "And Chiram constructed..." [Melachim I, 7.40-50].
Our teacher the Rema writes that "if a wedding fell on this Shabbos one should still read that of Chanukkah." Until here is the quote. He means to say that the Haftorah of "שוש אשיש" - "Rejoice greatly..." [Yeshayahu 61.10-62.5], which would normally be recited by a groom, is not said. This ruling applied to the custom his community had. We, however, never read a special Haftorah for a groom at any time.
When the first day of the month of Teves (Rosh Chodesh) occurs on Shabbos three Torah scolls are used. The first six readings of the Shabbos portion are read out of the first. In fact, it is prohibited to read more than six readings from the first Torah, as the seventh must be read from the second scroll in order to include one of required number of Shabbos readings. Though that be so, the actual portion read as the seventh reading is that of Rosh Chodesh, following the maxim that the more common occurrence has precedence. Even though Chanukkah has the relative advantage of having the status of a public commemoration of a miracle its commemoration will not be lost as the readings of the Maftir and the Haftorah are both related to Channukah. When reading the portion for Rosh Chodesh one begins from the words "וביום השבת" - "And on the day of Shabbos" since the day itself is Shabbos.
The Maftir is read from the third scroll and the Haftorah is that of Chanukkah, "רני ושמחי" - "Sing and rejoice".
If Rosh Chodesh Occurs on a weekday to Torah scroll are used. The fist is used for the three readings of Rosh Chodesh and the second scroll is used for the Chanuukah portion, according to whatever its portion is that day. (See further in chapter 685, section 4)
If the Torah reader mistakenly read part of the section of Rosh Chodesh for the fourth Aliyah there is no concern if a second Torah was not taken out, since Chanukka has no additional Musaf offering that would necessitate its own reading. This true even if the mistake was realized during the fourth reading, before it was completed - Chanukka's reading is considered lost. If, however, a second scroll was already removed from the ark, a fifth person must have a portion read on his behalf to avoid giving the appearance that the second scroll is somehow unfit to be used. This portion should be the Chanukka portion, and the Kadish is recited after this reading. If Kadish had already been recited after the fourth reading it need not be repeated (Magen Avraham).
Our teacher the Rema writes: "If one mistakenly began with the Chanukka reading he should stop at once and switch to the Rosh Chodesh reading." Until here is the quote. This would be comparable in its way to placing the head Tefillin before the arm Tefillin. (Magen Avraham)
Other dispute this ruling, directing the reader to complete the first reading, the Kohen portion, in the Chanukka section and only then switching to the Rosh Chodesh portion. (Taz and Rabbi N. Katz [WHO IS THIS?]). Either way, it should be clear that if three verses were already read from the Chanukka portion they count as one of the four Aliyot, even according to the Rema, so that only three more Aliyot are required to be read from the Rosh Chodesh section.
In terms of a practical ruling, one should follow the dissenting opinion since stopping in the middle of a reading would also create the false impression that there is defect (Pegam) in the first Torah scroll. The same practice would also be followed on a Shabbos Chanukka which fell on the day of Rosh Chodesh. (The fact that this law was stated in regard to the weekdays was simply to teach that even in this circumstance an additional 'Aliya' is required. The same would certainly apply to 'Shabbos' as well. (Rabbi Akiva Eiger))