Festival services “on one foot”:
The three Pilgrimage Festivals / Shalosh Regalim, when the Israelites traveled to Jerusalem, are Sukkot, Passover, and Shavuot. The two days of Shavuot, the first two and last two days of Passover, and the first two days of Sukkot plus Sh’mini Atzeret and Simchat Torah are considered “chag” (in Hebrew) or “yuntif” in Yiddish. They are like Shabbat, except that one may cook and carry. The middle 4 days of Passover and middle 5 days of Sukkot are considered “chol hamoed”, or “quasi-holiday”, because one eats matzah / in a sukkah respectively, but one may still go to work. Services on “chag” / “yuntif” are similar yet different to those on Shabbat.

Evening service
- No Kabbalat Shabbat because it’s not Shabbat
- If Shabbat is on a Festival, or if it is Shabbat Chol HaMoed, or if Shabbat directly follows a Festival, then we start Kabbalat Shabbat with Psalm 92 because we are already in the peaceful mood.t
- Ma’ariv / Arvit is same as Shabbat with a few changes
- The nusach tune is different, even though the words are generally the same, musically indicating that this is a different time. This is true even if it’s a Festival and Shabbat, because the Festival is less common.
- Mi Chamocha is sung to a different “signature melody” for each holiday
- Sukkot - Na’a’nu’im
- Pesach - Adir Hu
- Shavuot - Akdamut
- Instead of V’Shamru, we have a verse from Leviticus (23:44) about holidays (if Shabbat, do both)
- If Shabbat Chol HaMoed, we don’t do the Festival line
- The Amidah is still silent, but the middle part is about the festival instead of about Shabbat (or Shabbat is added to the Festival words if it’s both).
- If it’s Shabbat Chol HaMoed then the holiday is added to the Shabbat words.
- No Me’ein Sheva section unless Shabbat
- This section is added whether it’s Shabbat and Chag or Shabbat Chol HaMoed.
- Instead of the Friday evening Kiddush, we do the Festival evening Kiddush (with additions if it’s Shabbat)
- If it’s Shabbat Chol HaMoed, it’s just the regular Shabbat Kiddush.
- If it's the first 2 days of Passover, the Kiddush is not done during services because it will be done during the Seder.
- If it's the first 2 days of Sukkot, the Kiddush is not done during services but rather after services in the synagogue sukkah.
- Before Aleinu on Simchat Torah evening, a call and response prayer called “Ata Hareita” is done. Then the Torahs are taken out and 7 Hakafot of dancing are done before the Torah is read.
- Before Aleinu on Passover, starting on the second night, we count the Omer.
- Before the Mourner's Kaddish at the end, Psalm 27 is added on Sukkot (some also add on Sh'mini Atzeret).
- Big difference is the nusach (style of chanting the prayers that indicates the time of day and year).
- Even if it’s Shabbat and a Festival, we use Festival nusach
- If it’s Shabbat Chol HaMoed then we use Shabbat nusach

P’sukei D’Zimra
- It’s the same on Shabbat and Festivals, though some will add specific Psalms for various Festivals and take out the Psalms for Shabbat and Friday if it’s not Shabbat.

Shacharit
- Starts a few lines earlier for Festivals
- The beginning is chanted differently, indicating musically that we’re in Shabbat mode
- Not true for Shabbat Chol HaMoed
- Shacharit is otherwise the same, except that each prayer ends with a special musical ending that’s specific for Festivals. However, Eil Adon, along with the prayer before and after it, is not done unless it's Shabbat. Instead, HaMei-ir La'Aretz is added.
- The Amidah begins the same as Shabbat, both with words and tune, but the Kedushah switches to Festival nusach before L’Dor Vador
- There is a different set of words for the middle of the Amidah, talking about the Festival rather than Shabbat. Certain paragraphs have “insert name and nickname of Festival here” lines.
- If it’s Shabbat and a Festival, there are extra words added to the Festival Amidah.
- If it’s Chol HaMoed and Shabbat, then the “Ya’aleh V’yavo” paragraph is added to the Shabbat Amidah.
- The Kaddish Shalem at the end of the Shacharit section (after Hallel) is chanted the same whether it is Chag or Shabbat.

Hallel
- After the Amidah and before the Kaddish Shalem, Hallel is added whether it is a “yuntif” day or Chol HaMoed (tefillin is removed first on Chol HaMoed). It’s also added on Chanukah, Rosh Chodesh, and Yom HaAtzmaut, though some don’t say the blessing for that last one.
- Hallel is a section of the service primarily comprised of Psalms 113 through 118.
- It is preceded by a blessing, which the leader says and the congregation repeats.
- On Sukkot, before we start Hallel we say a blessing and then wave the lulav and etrog. We also wave it to the Na’anu’im tune during “Hodu Ladoshem” and “Ana Adoshem”.”Hodu Hadoshem” can also be done to “Adir Hu” on Passover.
- On the last 6 days of Passover and Rosh Chodesh, we only do a “half Hallel”, meaning that we don’t do the first half of Psalms 115 and 116. Originally, Hallel wasn’t done in the Temple on the last 6 days of Passover since there weren’t unique sacrifices then, unlike on Sukkot. The Jews of Babylonia added it then, but made it shorter to show that it wasn’t originally done.
- A later explanation why we don’t do the full Hallel after the beginning of Passover is that we are sad that others had to suffer so we could be free.

Torah Service
- On Simchat Torah, the Ata Hareita and Hakafot supercedes the regular Torah service. After the Hakafot, continue with Shema and Echad, except that the Echad line also has “v’nora” in it.
- If it’s not Shabbat, after the Ark is opened we add the Thirteen Attributes 3 times, followed by “Yih’yu L’ratzon” and then “Va-ani t’filati”
- On the other hand, “Bei ana racheitz” is only done on Shabbat.
- Two Torahs are taken out (or 3 for reading on Simchat Torah, plus all the other ones for dancing).
- There are only 5 aliyot plus Maftir on a Festival, unless the Festival is also Shabbat in which case there are 7 plus Maftir.
- There are 4 aliyot on Chol HaMoed, unless it’s also Shabbat.
- Before the Torah reading on the first day of Shavuot, there’s a mystical poem recited called “Akdamut”.
- The Torah reading on the first day of Shavuot is done using the "Upper Trope" ("Ta'amei Elyon") for the Ten Commandments.
- The Maftir is always related to the holiday (specifically the sacrifice in Numbers 28-29) and it comes from the second Torah.
- The Haftarah is specific to the holiday, even during Chol HaMoed Shabbat.
- The Blessing After the Haftarah has a different last paragraph if it’s a Festival (with additions to the Festival paragraph for Shabbat).
- On Shabbat Chol HaMoed, we add Shabbat language into the Festival words on Sukkot, but we just do the regular Shabbat version on Passover.
- While it’s unclear why this is so, it may be connected to the downplaying of Passover with only a Half Hallel during the last 6 days.
- It might also be connected to the fact that in the Bible more sacrifices were offered on Sukkot.
- There is no Y'kum Purkan if it's not Shabbat.
- When taking back the Torah, if it’s Shabbat Psalm 29 is said, but if it’s not Shabbat then Psalm 24 is said.

Megillot
- Different communities do them at different times, but each holiday has a special book of the Bible chanted.
- While most people think of Esther when they think of “The Megillah”, there are actually 5 megillot, or scrolls, each of which is chanted at a specific time throughout the year.
- Esther is chanted on Purim, using unique trope for Esther.
- Eicha / Lamentations is chanted on Tisha B’Av, using sad trope that is unique for Eicha.
- The other three books are all chanted using the same trope, called “RaKaSh” as an acronym.
- Ruth on Shavuot Day 2
- Kohelet / Ecclesiastes on Shabbat Chol HaMoed Sukkot
- Shir HaShirim / Song of Songs on Shabbat Chol HaMoed
Passover

Yizkor
- There is an extra section added to the service on the last day of Passover, the last day of Shavuot, on Sh’mini Atzeret at the end of Sukkot, and on Yom Kippur at the end of the High Holidays. It’s called “Yizkor”, having to do with remembering.
- The idea is that we are probably remembering our loved ones at the holidays anyway, so this gives us a container for those memories.
- Different communities do Yizkor at different times, but it is generally some time between the end of the Haftarah and the end of the service.
- While some people leave the room if their parents are alive, many people will stay for one or more of the following reasons:
- To remember other loved ones
- To say Yizkor for those who died with nobody to remember them, particularly but not exclusively for those who died in the Holocaust
- To be a supportive presence for others who are remembering loved ones
- The sermon on these holidays often leads into Yizkor.
- Yizkor starts with a few prayers together, and then people are left to their private prayers and memories.
- The shaliach tzibur, prayer leader, then brings everybody together for “Eil Maleh” prayers said on behalf of all those who died and/or those who died in the community and/or those who died in the Holocaust / as martyrs and/or defending Israel.
- Often Psalm 23 is chanted, and then Mourner’s Kaddish is done.
- In communities where Yizkor ends the entire service, sometimes “Oseh Shalom” is sung at the very end to serve as a musical transition to Kiddush lunch.

Musaf
- Like Shacharit, Musaf usually starts the same on Festivals as on Shabbat, and then transitions to a different nusach and different set of words after the Kedushah.
- The Musaf Kedushah has an extra line, “Adir Adireinu”, toward the end.
- The Chatzi Kaddish going into Musaf and the Avot paragraph at the beginning of the Amidah is sung using High Holiday nusach on Sh’mini Atzeret and the first day of Passover.
- On Sh’mini Atzeret the G’vurot paragraph is replaced with “Geshem”, praying for rain in Israel.
- On the first day of Passover, the G’vurot paragraph is replaced with “Tal”, praying for dew in Israel.
- The Shaliach Tzibur traditionally will wear a white kittel robe on these days.
- At the end of the Amidah on all days of Sukkot, Hoshanot are added, processing around the room with lulav and etrog.
- On Shabbat during Sukkot we do Hoshanot but without lulav and etrog.
- Before the Mourner's Kaddish at the end, Psalm 27 is added on Sukkot (some also add on Sh'mini Atzeret).

Mincha
- There is no special Mincha for Festivals. However, Uva L’Tziyon is added after Ashrei, and Ya’aleh V’yavo is added to the Amidah.
- When Festival Mincha is on Shabbat, Tzidkatcha is skipped.

Ma’ariv / Arvit
- It’s a regular weekday evening service. However, the Ata Chonantanu paragraph is added to the Amidah in the 4th blessing. If you are going into Chol HaMoed, then Ya’aleh v’Yavo is added to the Amidah as well.
- On Sukkot, Psalm 27 is added before the Mourner’s Kaddish.
- Havdalah (done in the sukkah at the end of the beginning of Sukkot) only has the blessing for grape juice / wine, and even if you are going into Chol HaMoed then it ends with “haMavdil bein kodesh l’chol”.

Appendix: Adir Adireinu
According to Rabbi Jonah Rank
The history is that Addir Addirenu used to be recited on both Shabbat and all days of Yom Tov. To my knowledge, chol ha-mo'ed never prompted its recital. The communities that would want to recite it on the Shabbat between the beginning and end of yamim tovim, depending on a variety of factors, could opt to recite it on Shabbat Chol Ha-Mo'ed inasmuch as it is also Shabbat like any other Shabbat. But it has largely (though not entirely) fallen out of practice to recite it at all on Shabbat, even when Yom Tov coincides with Shabbat. (Note that ArtScroll and Koren have disagreed on whether to recite it when Yom Tov and Shabbat coincide.)

https://zeramim.org/past-issues/volume-issue-3/rank-addir-addirenu-shabbat-beyond/#_edn130 has one of a few versions of my article on this.