All the holidays and festivals change dates, for us, each year... the Jewish New Year, Simchot Torah, Chanukah, Tu B'Shevat, Purim, Passover... etc. That's because as you said, it is the Jewish calendar that is followed.
If you follow the Jewish calendar. ...the holidays are actually on the same date each year... Chanukah is always on the 25th of Kislev, for example. But since we don't use that calendar, but use the Gregorian calendar, which follows the solar year, the dates change all the time, here in the Western world.
So.... why does Rosh Hashanah start in September? That's when the "New Year" starts... Rosh Hashanah means "the first of the year." It's celebrated on the first and second day of Tishri. (It can be late certain years, and fall on a day in October too.)