The Jewish New Year initiates ten days of repentance that culminate with the Day of Atonement. The Torah does not refer to it as Rosh Hashonah (that name first occurs in the Talmud), but as the Memorial of the Blast of the Horn (Zichron Teruah) or the Day of the Blast of the Horn (Yom Teruah). Traditionally, Rosh Hashonah is associated with the Birth of the World.
The Jewish calendar is lunar, which means the 1st day of Tishrei can occur anywhere from the beginning of September to the beginning of October, depending on the year of the Jewish calendar.
Traditional Rosh Hoshonah foods include apples and honey (for a sweet year) and a round challah (egg bread) signifying the cyclicality of a year.
Rosh Hashonah citations in the Torah and Talmud include:
Leviticus 23:24 - Speak to the Children of Israel, saying: in the seventh month on day one of the month there will be for you a Shabbaton [a Sabbath observance], a memorial of the blast of a horn, a solemn assembly.
Numbers 29:1 - And in the seventh month on day one of the month there will be a holy convocation for you; all the work of the service you will not do, it is a day of the blast of the ram's horn for you.
Tractate Rosh Hashonah of the Babylonian Talmud provides the first reference to the 1st day of the 7th month as the New Year. It also connects Rosh Hashonah with the repentance associated with Yom Kippur (the Day of Atonement)
Gedaliah was the governor of Judah appointed by Nebuchadnezzar who was assassinated shortly after the fall of the First Temple. This is considered a minor holiday, observed by a fast that begins at sunrise and ends at sunset on the same day (most Jewish holidays begin the night before; this is an exception). The details of Gedaliah's assassination are described in 2 Kings 25:25-26 and Jeremiah 41.
The Day of Atonement is considered the holiest day of the Jewish calendar. According to tradition, it is the day on which G-d makes and seals the final decision on who will live and who will die in the coming year.
In traditional practice, this is a fast day, with prayer and repentance for 24 hours, on the 10th day of the 7th month (Tishrei 10) ending with the visibility of the third star in the sky.
The Jewish calendar is lunar, which means the 10th day of Tishrei can occur anywhere from beginning of September to the beginning of October, depending on the year of the Jewish calendar.
Although the holiday is referred to as Yom Kippur (Day of Atonement), in the Torah, it is called Yom Kippurim (Day of Atonements).
The Yom Kippur citations in the Torah and Talmud are:
Leviticus 16:29 - "And it will be for you as a law for eternity; in the seventh month, on the tenth of the month, you will afflict your souls, and all the work you will not do, the native or the stranger who lives in your midst."
Leviticus 23:27 - "Also, on the tenth of this seventh month, a Day of Atonements, a holy gather it will be to you, and you will afflict your souls, and you will make an offering of fire to YHVH."
Leviticus 25:9 - "And you will bring a ram's horn to sound in the seventh month on the tenth of the month on the Day of Atonements; you will bring the ram's horn through all your land."
Number 29:7 - And on the tenth of the month, in the seventh [month] it will be a holy assembly for you, and you afflict your souls; all the work you will not do."
Tractate Yoma of the Babylonian Talmud includes the rules and ordinances for the Day of Atonement, include the animal sacrifices (a practice no longer performed).