The Jewish year 5775 begins at sundown tonight (Wednesday). In Hebrew, “new year” is r'osh ha-shana: literally, “head [of] the year.” Interestingly, the Arabic term for “new year” is the same: r'as as-sana. Clearly this expression goes back a long, long way, possibly even to Proto-Semitic times. In any event, the phrase long predates monotheism. One should probably posit an Arabic—possibly Moorish—origin for the Italian word for “new year,” capodanno. Three guesses what that means literally.
“New Moon” in Hebrew is r'osh hodesh, literally (you guessed it) “head of the month.” Why would the head of something come to mean its beginning?
I can think of two possibilities.