“G-d Spoke to Moshe saying: Speak to the children of Israel and say to them: G-d’s appointed festivals that you are to designate as special times, these are My appointed festivals.” (Lev. Ch.23 V.1-2) The Torah then proceeds to mention all the festivals: First Shabbos is mentioned, followed by Passover, Festival of Weeks (Shavuos), Rosh Hashanah, Yom Kippur, Sukkos, and Shmini Atzeres.
There are two things which need to be explained here: 1. What is the purpose of festivals, especially since we already have the concept of Shabbos? 2. What does it mean when it says in the verse “that you are to designate”?
When G-d created the world He did all the physical acts of creation in six days. He then “rested” from creating on the seventh day. He did not rest because of tiredness; rather, He set aside a day for spirituality. When His chosen people (chosen because they chose to accept the responsibility of keeping the Torah) observe the Shabbos, they acknowledge that not only did G-d create the world, He created for a greater purpose than merely creating or performing physical acts.
Shabbos is a day that He set aside for us, to acknowledge the very reason that we were created: to forge a relationship with our creator.
The holidays are the days that we learn how to forge this relationship with Him. Many of the most important ideas of Judaism are learned on the holidays, such as: gratitude (Passover; for G-d having taken us out of Egypt), observance of the Torah, and G-d speaking to us directly (Shavuos, when we received the Torah at Mount Sinai and G-d spoke to us), and trust in G-d (Sukkos; when we leave our secure home and go in the Sukkah). We also learn on the holidays the concept of renewal (Rosh Hashanah), and to seek atonement from G-d and our fellow man (Yom Kippur). We also learn about the intense love G-d has for us (Shmini Atzeres; the extra day added on after Sukkos, because G-d could not bear for the holiday season to end).
How do we internalize these important messages? That is what it means in the aforementioned verse “that you are to designate”.
Shabbos happens every seventh day. The holidays happened on specific days of the month. The months did not come in on their own (see Parshas Pinchas). Whether a month has 29 or 30 days is contingent not only on the testimony of Jewish people as to the position of the moon, but also on the court actually sanctifying the month (Tractate Rosh Hashanah 25b).
Shabbos is put into a place by G-d and is thus Heavenly influence from above, and by keeping Shabbos we receive that influence. So our keeping it creates a flow between us and G-d. The holiness of the holidays comes from us (the Jewish people), thus returning that flow. It is through these two types of spiritual influence (initiated by G-d, or initiated by us) that the cyclical nature of our relationship with G-d is demonstrated.