What was the order of "Shacharit" in the Bet Hamikdash?

Question

I'm assuming that the majority of the format of the Siddur that we currently have and use did not exist during the time of the 1st Bet Hamikdash. Perhaps, by the 2nd Bet Hamikdash, some items were added. I'm assuming that the only two items from the Siddur that were part of Shachrit were:

• Shema
• Song of the Day

I'm assuming that these two items "surrounded" the offering of the morning tamid offering. When were these two items said (Levi'im sang the daily song)?

My assumptions may be wrong. If there were any items currently in the Siddur / Shachrit service that were added, please state what they are and when they were said repective to the tamid.

Answer 1

After the korban tamid was slaughtered, skinned and butchered, its blood thrown on the altar and its limbs placed on the altar's ramp, they would gather in the lishkat hagazit and recite:

• Ahavat Olam
• Aseret HaDibrot
• The three paragraphs of Shema
• Emet v'Yatziv
• R'tzei
• Sim Shalom

וְאַחַר שֶׁמַּעֲלִין הָאֵיבָרִים לַכֶּבֶשׁ מִתְכַּנְּסִין כֻּלָּן לְלִשְׁכַּת הַגָּזִית. וְהַמְמֻנֶּה אוֹמֵר לָהֶם בָּרְכוּ בְּרָכָה אַחַת. וְהֵן פּוֹתְחִין וְקוֹרִין אַהֲבַת עוֹלָם וַעֲשֶׂרֶת הַדִּבְּרוֹת וּשְׁמַע וְהָיָה אִם שָׁמוֹעַ וַיֹּאמֶר וֶאֱמֶת וְיַצִּיב וּרְצֵה וְשִׂים שָׁלוֹם.‏

(Rambam, Hilchot Temidim u'Musafim 6:4)

The ketoret was then offered, the remaining two lamps of the menorah prepared, and the limbs of the tamid were taken from the ramp and offered up on top of the altar, after which the kohanim recited birkat kohanim.

(Rambam, Hilchot Temidim u'Musafim 6:4-5)

They would then offer the flour-offering which accompanied the tamid, the kohen gadol's flour-offering, and then the wine libation. During the wine libation, the levi'im would recite that day's song, to musical accompaniment.

(Rambam, Hilchot Temidim u'Musafim 6:5)

Answer 2

Joel K's answer above covers the larger-scale system pretty accurately. But the text of the prayers themselves definitely went through some changes, and the clearest come from the רצה. Obviously, the modern text we have for רצה wouldn't have worked when the Temple was standing, considering its repeated references to the destruction of the Temple and the exile. In My People's Prayer Book, volume 2 (pp. 161–162), Rabbi Lawrence A. Hoffman reconstructs the text that would have been used in the Temple era thusly:

1. First he removes any mentions of the destruction – removing the phrases וְהָשֵׁב הָעֲבוֹדָה לִדְבִיר בֵּיתֶךָ and וְתֶחֱזֶינָה עֵינֵינוּ בְּשׁוּבְךָ לְצִיּוֹן בְּרַחֲמִים, as well as the chatimah of הַמַּחֲזִיר שְׁכִינָתוֹ לְצִיּוֹן.
2. Then he suggests that the addition וּתְפִלָּתָם would have postdated the destruction, reflecting the unique Rabbinic emphasis on prayer as a means of "replacing" sacrifice, and that בְאַהֲבָה would have been added at the same time since בְּרָצוֹן would have covered the acceptance of sacrifices (being oft associated with the sacrificial rites).
3. Then he uses the old Eretz Yisrael rite chatimah (as recorded in the Cairo Geniza and still used among some Ashkenazim on Musaf before dukhening), שֶׁאוֹתְךָ לְבַדְּךָ בְּיִרְאָה נַעֲבוֹד, since the present tense would have been accurate in Temple times.

So he ends up with a much shorter רצה prayer – רְצֵה ה׳ אֱלֹהֵינוּ בְּעַמְּךָ יִשְׂרָאֵל, וְאִשֵּׁי יִשְׂרָאֵל תְּקַבֵּל בְּרָצוֹן, וּתְהִי לְרָצוֹן תָּמִיד עֲבוֹדַת יִשְׂרָאֵל עַמֶּךָ.בָּרוּךְ אַתָּה ה', שֶׁאוֹתְךָ לְבַדְּךָ בְּיִרְאָה נַעֲבוֹד. Which he then translates as "Find favor, Ad--ai my God, in your People Israel. And the fire-offerings of Israel, accept favorably. And may you find favor in the tamid, the sacrificial service of Israel your People. Blessed are You, Ad--ai, whom we serve with awe."

Note his suggestion that the תָּמִיד of the final clause referred to the tamid-offering, which the service surrounded. In a sense, it's the prayer of prayers in this rite. Unlike in a modern Amidah, the רצה would have almost certainly been the spiritual high point of the service – the key petitionary text.

Also, the wine-libation service is listed in Tamid as well, including the psalms of the day.

1. They would give the kohen the wine.
2. Two kohanim with trumpets would blast, tekiah, teruah, tekiah.
3. The kohen would bend down to pour the wine.
4. The cymbal would sound, flags would be waved, and the Levites would sing psalms.
5. As each psalm ended, the trumpets would blast and the people would prostrate themselves.

That pretty clearly implies that more than one psalm would be sang per day. Though the psalms of the day are listed in Tamid 7:4, it seems unlikely that only those seven psalms would be sang as we do today. In the minor tractate Sofrim it lists a series of psalms that were read for each major holiday, more than one in every case. Probably different psalms would be sung each day, including but NOT limited to the seven of the week.