Here is the official description of the Jewish (agricultural) New Year (Rosh HaShanah), a day that became centrally significant only over time:
Num. 29:1   On the first day of the seventh month you shall have a holy convocation; you shall not work at your occupations. It is a day for you to blow the trumpets, 2 and you shall offer a burnt offering, a pleasing odor to the Lord: one young bull, one ram, seven male lambs a year old without blemish. 3 Their grain offering shall be of choice flour mixed with oil, three-tenths of one ephah for the bull, two-tenths for the ram, 4 and one-tenth for each of the seven lambs; 5 with one male goat for a sin offering, to make atonement for you. 6 These are in addition to the burnt offering of the new moon and its grain offering, and the regular burnt offering and its grain offering, and their drink offerings, according to the ordinance for them, a pleasing odor, an offering by fire to the LORD.
1. A day of rest — celebrated at home (for Diaspora Jews) unless one lived in Jerusalem.
2. A day of blowing the shofar (at the Temple): this was distinct to this day. It signalled a time to reflect on one’s life and begin anew. Over time the shofar was blown also in local synagogues and became connected to a day of judging sin (cf. Isa. 58:1; Ps. 47:5-6; 81:4; 89:15-16).
3. A day of offering to the Lord.
4. A day of atonement.
The development of Israel’s calendar is beyond the scope of this brief post. Just compare Exodus 12:1 and 23:16, which seems to indicate an original new year event in Tishri and then later the first month being Nisan. See also Ezk 40:1. The Babylonian calendar played its role as well.