1) Birkot Hashachar:

The introductory morning prayer offers thanks for basic gifts such as sight, Torah, and life.

2) Pesukei d’Zimra:

The opening “verses of praise” come from psalms and other poems and serve as a spiritual “warm-up” for the rest of the service of praise and thanksgiving.
The morning service begins with the prayer “Baruch She’amar v’hayah ha’olam” – “Blessed is the One who spoke and the world came into being.”Pesukei d’Zimra concludes with the prayer “Yishtabach shimcha” – “May Your name be magnified,” and the Hazi Kaddish, which divides this section of the service from the next.

3) Bar’chu:

The central part of the service begins with this call to public worship. The congregation stands. The leader recites, “Bar’chu et Adonai, hamevorach,” – “Blessed is Adonai, Who is to be blessed for all time.” The leader repeats this line.

4) Shema:

The Shema proclaims the oneness of God and affirms our commitment to loving and following God. There are two blessings before it, one acknowledging God as creator of light and as the source of the laws of nature in general, and one acknowledging His love for us and His giving us the Torah. The Shefma is followed by a blessing that recounts the miraculous redemption of the Jewish people in the past.

5) Amidah:

The Amidah is the central prayer of all Jewish prayer services. The Amidah is said silently, while standing. Themes include God’s relationship with the patriarchs [and matriarchs in some congregations] and with the Jews through history, His power over nature, His sanctity and Kingship, the sanctity of the holiday, our desire to worship him properly, thanksgiving for life, and the desire for peace. Several special inserts into the Amidah for the High Holy Days repeatedly express our yearning for the gift of life. The paragraph at the end of the Amidah represents an opportunity to add individual reflections and to offer individual prayers.

6) Amidah Repetition:

After the congregation has finished reciting the silent Amidah, the leader recites aloud an expanded version of the prayer. In the third blessing, we say the kedushah (sanctification of God), sung by the leader and the congregation, with the congregation standing at their places. Other additions to this version are poems stressing the themes of God’s monarchy, remembrance, redemption, and the themes of judgment, repentance, righteousness, and exaltation of the divine. During certain key passages, the Ark is opened, and the congregation rises out of respect.

7) Avinu Malkeinu:

In the “Avinu Malkeinu” prayer, we speak to God as a parent and sovereign. The prayer consists of requests for compassion, life, and happiness.

8) Torah service:

The Torah service begins with the opening of the ark and the removal of the Torah. At this point, the leader carries the Torah through the congregation while singing.
The reading for the first day of Rosh Hashanah begins by saying that “God visited Sarah.” The theme is that God has steadfastly kept in mind Sarah’s distress over her childlessness, and that He keeps His promise to relieve that distress. We read it on Rosh Hashanah because of our focus on God as the One who remembers us and redeems us.
The reading for the second day tells the story of the binding of Isaac, in which Abraham is willing to sacrifice his son at God’s command.
On both days there is an additional reading, the maftir, containing biblical instructions about the sacrifices that were offered in the Temple on Rosh Hashanah. The Torah reading is split into five sections, or aliyot, plus the maftir. (Seven aliyot plus maftir are read on Shabbat.)
The Torah service also includes a Haftarah, a reading from the Prophets. The Haftarah for the first day tells the story of Hannah, whose distress over the childlessness parallels Sarah’s; the story tells us that God “remembered” Hannah. On the second day, the Haftarah, from Jeremiah, recounts God’s promise to “remember” Ephraim and redeem the Jewish people.
The Torah service concludes with the return of the Torah scrolls to the ark.

9) Shofar:

The Shofar is traditionally sounded 100 times during the service. (The Shofar is not blown on Shabbat.) The first 30 blasts come now. There are three types of Shofar blasts – tekiah, shevarim, and teruah. The tekiah (a long blast) is linked with crying out in distress; the shevarim (three shorter blasts) and the teruah (nine or more staccato blasts) are linked, respectively, with moaning and sobbing. Before each blast, the leader of the services or another member of the congregation calls out the name of the blast to come.

10) Musaf:

Musaf, the additional service for Shabbat and holidays, commemorates the extra sacrifice offered on these days during the time when the Temple stood in Jerusalem. The beginning and end of the Musaf Amidah are virtually identical to the beginning and end of the Amidah for the evening and morning services. In the middle, however, are three sections unique to this day. Each section focuses on one of the three key themes of the day, and each includes 10 biblical verses mentioning that theme. The three themes are:

Malchuyot (verses of sovereignty): God as sovereign.

Zichronot (verses of remembrance): God’s mindfulness of us and our situations.

Shofarot (verses of the Shofar): God as redeemer.
During the silent Amidah of Musaf, the rabbi (or another member of the congregation) will blow the shofar 10 times at the end of each of these sections.

11) Musaf Amidah Repetition:

The leader’s repetition of the Amidah follows the same structure as the silent Amidah, though it is expanded by the addition of a number of poems. Preceding the recitation of the Amidah is the singing of the “Hineni” prayer, in which the leader asks for the ability to be an acceptable and effective representative for the congregation.
Key moments in the repetition of the Amidah:
Unetaneh Tokef: This prayer proclaims the importance of Rosh Hashanah as the day on which judgment is passed. According to the prayer, it is decided on Rosh Hashanah “who shall live and who shall die, who shall live a full life and who shall not.”
Kedushah: In the third blessing, we add the kedushah (sanctification of God), sung by the leader and the congregation, with the congregation standing at their places.
Aleinu: This prayer, which has now become one of the final prayers of all Jewish services, began as part of the High Holiday Amidah. During this prayer, congregants bow, but during the high holidays some people kneel on the floor in a re-enactment of the way that Jews worshipped in the Temple.
Shofar: The shofar is blown 20 times at the end of each of the “Malchuyot,” “Zichronot,” and “Shofarot” sections of the repetition.
Priestly blessing: Toward the end of the repetition, the Kohanim (men descended from Aaron) perform the priestly blessing (“May the Lord bless you and keep you”).

12) Concluding prayers:

The service ends with the Kaddish Shalem (plus an additional 10 shofar blasts to complete the total of 100 for the day), Aleinu, Mourner’s Kaddish, and the singing of Adon Olam.

Source: Ohr Kodesh Congregation in Chevy Chase, MD

Rosh HaShanah: Guide to the Morning Service