In Exodus 12:1-2, Yahweh tells Moshe that the year from then on will begin "this month", but it is not for the purpose of counting years; rather, it is in commemoration of the momentous transformation, from slavery to physical liberation from Egypt.
Passover, on the other hand (which many call the beginning of the New Year) is celebrated in its own right as the pivotal event which led to the Exodus. Therefore, the 1st day of Nisan became the date for the first month of the Hebrew/Jewish calendar.
In other words, months in the Hebrew/Jewish calendar are numbered beginning with the month of Nisan as explicitly stated in the Torah. In fact, the title "First of the Months" ("Rosh Hodashim" in Hebrew) is reserved in the Torah for the month of Nisan (Exodus 12:2).
Then there is also the "civil year", the point in the year from which the years are counted. This is the historical date agreed upon by the sages that was the date of the creation of man. Yom Teruah/Rosh Hashana (in September 2010), kicked off the year 5771 from the creation of Adam.
Counting years from Tishri (Yom Teruah/Rosh Hashana) is also the date for calculating the release year (i.e., the "Sh'mittah" year, which means "Sabbatical" year, which is every 7th year in the 49-year cycle that governed the Kingdom of Israel [10th century B.C.E. to 8th century B.C.E.] and Kingdom of Judah [10th century B.C.E. to 6th century B.C.E.] in biblical times), and the date for calculating the Jubilee year (a Jubilee year or "Yovel" year in Hebrew is the year after the 49-year cycle that governed the Kingdom of Israel and Kingdom of Judah in biblical times, i.e., the 50th year).
The 1st day of Tishri was also the date that determined the beginning of the year when it came to the three years that the fruit of a tree must be left unpicked (Leviticus 19:23). The 1st day of Tishri was also the date for the "tithe of crops" for the Levites and the Priesthood ("Cohanim" in Hebrew), whose dedication to holy service prevented them from working on the land like the other Hebrews.
For the “rest of the story” please check out our indepth article!