At Odds with Hillel
I believe that the pre-calculated Hillel II calendar used in Judaism is incorrect, and that the Bible prescribes an observance-based system for determining the beginnings of months. Hillel II was introduced in the fourth century A.D., in the wake of the Diaspora, as a way of ensuring that Jews living in places remote from the Sanhedrin could nonetheless celebrate the feasts on the same day. This tactic was useful for maintaining unity, but not useful for proper adherence to Adonai's timekeeping system. There are technical problems with the Hillel II system (as even the Jewish leaders will admit).
Proper reckoning of the appointed times requires actually viewing with the naked eye the first sliver of the waxing moon. This sighting must be done from near Jerusalem, in the Promised Land, which is the sacred center of the earth.
Thanks to the internet, anyone in the world can adopt this method of counting months based on Israeli lunar sliver sightings. There are several websites which report the sighting of the new moons from Israel, and some of them also predict (based upon astronomical computer models) the beginnings of future months.
The Gregorian Dates
Therefore, as you plan to take off of work on the High Sabbaths, make sure you request to have off work on the dates following the listed Gregorian dates.
Here are this year's start dates for the mo'ediym:
9/30: Yom Kippur (Day of Atonement)
10/5: First day of Sukkot* (Feast of Tabernacles)
10/12: Shemini Atzeret (The Last Great Day or The Eighth Day of Assembly)
*Sukkot lasts for seven days, but only the first day is a High Sabbath