The main thing to know about Tishri is that it’s a month with many holy days: Rosh HaShanah, the Days of Awe, and Yom Kippur, followed by the week-long Feast of Sukkot, Shemini Atzeret, and Simchat Torah. (Click the links for more info on those holidays.)
There is also a minor fast day, Tzom Gedaliah, which memorializes the death of a righteous governor of Judea in the years after the destruction of the Temple by the Babylonians. Read 2 Kings 25: 24-25 or Jeremiah 41 for the story.
It’s a busy month!