Rosh Hashanah falls on the first of Tishrei, and Tishrei is the seventh month. The Torah counts Nisan as the first month, so the Jewish new year is celebrated at the start of the seventh, which looks like an error and is not.
Rabbinic tradition resolves it by recognising four new years for different purposes: the first of Nisan for kings and festivals, the first of Elul for the tithe of cattle, the first of Tishrei for years, sabbatical cycles and jubilees, and the fifteenth of Shevat for trees. This is a calendar with multiple year-starts serving different functions, which is not unusual in the ancient world and is not unusual now, fiscal, academic and civil years routinely differ.
The Torah does not call the day a new year. Leviticus and Numbers name the first of the seventh month as a day of rest, of memorial proclaimed with blast of horns, and of holy convocation. The name Rosh Hashanah, and the day's character as the anniversary of creation and the day of judgement, are rabbinic developments, and the Mishnah's discussion is where they are set out.
The shofar is the day's defining act. It is a ram's horn, and the sounds are prescribed: a long blast, a broken one, a series of short ones, and a very long one, in fixed sequences totalling a hundred notes in most customs. The obligation is to hear it, not to blow it.
The day begins the ten days of repentance that end at the Day of Atonement, and the liturgical language throughout is judicial: books opened, a decree written, and the possibility of it being changed.