Ezra - Chapter 3
The month of Tishrei rolled around, replete with holidays, so the people all traveled to Jerusalem. (Rosh Hashana, Yom Kippur, Succos and Shemini Atzeres all fall in Tishrei.) Yeshua the High Priest got up with the other kohanim, Zerubavel (the prince) and other nobles, and they built an altar in the proper place, upon which to offer sacrifices. (Apparently, it was permissible to do so in the proper location even in the absence of a rebuilt Temple.) The Jews were concerned that the locals, who had been living there since the exile, might interfere with their plans to rebuild the Temple, so they were quick to establish their presence there, as well as the fact that their actions were in accordance with the king's wishes. This altar helped them establish their credentials (in addition to its obvious, primary purpose of re-establishing the rapport with G-d).
The Jews resumed the daily morning and afternoon sacrifies and they celebrated the holiday of Succos, which has a different number of sacrifices each day. After Succos, they continued by offering all the Rosh Chodesh ("new moon") and holiday sacrifices in their proper times. They also brought offerings that the people volunteered to G-d. They started sporadically on Rosh Hashana, but the foundation of the Temple had not been laid; by Succos, they were in full swing.
The people paid workmen to bring cedar wood from Lebanon, just as when the first Temple was built. In the second year of the Jews' return, Zerubavel, Yeshua, the rest of the leaders, the priests and Levites, and all the rest gathered for the groundbreaking. The Levites, age 20 and up, sang during the construction, just as they used to in the Temple. Yeshua and others, including Levite minors, joined in the singing. (The Talmud in Arachin 13b says that, while Levites under the age of 20 could not perform other Temple functions of their Tribe, they could join in the singing.) The builders laid the foundation of the Temple and the kohanim, in "uniform," took their places, while trumpets were sounded. The Levites played cymbals and sang the Psalms that David had composed for them. They sang responsively, with great enthusiasm and gratitude to G-d for all the goodness He had given them. (They sang, "He is good and His kindness is eternal..." as per Psalm 118.) The crowd went wild with rejoicing, though the oldest among them wept bitterly because they remembered the first Temple and knew what they were missing. Some of the elders, who had not seen the original Temple, rejoiced, but they were outnumbered by the elders who cried, who could be heard from far away.