The story of Chanukah relates how Antiochus and the Syrian Greeks tried to prevent the Jewish people from observing their religion. To this end, they prohibited them from performing three major Mitzvot: Shabbat, Rosh Chodesh, and Brit Milah. What do these Mitzvot have in common and why did Antiochus want them stopped? They all are core elements of Judaism in their own way, so they had to be stopped in order to make Jews abandon their religion.
Shabbat symbolizes that Hashem created the world in six days and rested on the seventh. By observing Shabbat, we proclaim to the world that Hashem is the Creator and that none is higher than He. The Greeks, however, believed that they were the highest power in the universe, so they didn't allow Jews to observe Shabbat.
The observance of Rosh Chodesh was prohibited because it is the cornerstone that enables the Jews to observe all other Jewish holidays. When the Sanhedrin would declare Rosh Chodesh each month, the dates of all holidays that would fall in that month automatically were set. For example, the date of Pesach can be determined only based on Rosh Chodesh Nissan. Since the Jewish holidays strengthen the relationship between Hashem and the Jews, Antiochus wanted to prohibit the observance of all holidays. The most efficient way to do this was to ban Rosh Chodesh.
Finally, the Mitzvah of Brit Milah demonstrates the connection between the physical and spiritual. The Brit is the physical mark that a person is Jewish and symbolizes his spiritual connection to Hashem. For this reason, Antiochus had to stop Milah.
Each of these three things also has a connection to Chanukah. Every Chanukah contains at least one Shabbat and a Rosh Chodesh. Also, Chanukah contains eight days, the same number of days of a baby's life before his Brit Milah. Even though Antiochus tried to force us away from our religion, we still stayed strong and kept it. Because of the Jews' perseverance in the time of the Yevanim, Jews are able to exist today.