The three festivals Pesach, Shavu'ot, and Sukkot are called the pilgrimage festivals because, while the Temple stood, all Jewish men were commanded to go up to Jerusalem to celebrate the festivals there, and to offer special sacrifices, and the public Musaf" ("additional") sacrifices for each of these festive days.
These three festivals are days of exceptional joy that have a special positive commandment, "and you shall rejoice on your festivals" (Deuteronomy 16:14).
In a more spiritual sense, they are celebrations of God's Divine manifestation to His people in the Temple.
The festivals were, in this sense, "appointments" between God and Israel, as it is written, "all your males shall appear before the Lord God" (Exodus 23:17).
The order of these three festivals in the Torah is always Pesach, Shavu'ot, and Sukkot.
Besides the fact that this is in accordance with the calendar year beginning with Nisan (which also applies to various laws concerning the festivals), there is also an inner significance to this order.
On one hand, there is a triad of attributes: Mercy, Strength, and Glory.
Mercy is for the Pesach festival, which concerns redemption.
Strength is for the Giving of the Torah, which prescribes the laws and edicts.
Glory is for Sukkot, which is a total manifestation of joy.
In addition, there is the historical significance of the festivals:
On Pesach–the first formation of nationhood in the exodus from Egypt.
On Shavu'ot–the granting of true meaning and purpose for the redemption by the Giving of the Law.
On Sukkot–completion and repose.
There is also the order of the seasons in the year:
Pesach is the festival of spring, the time of flowering, when the grain begins to ripen.
Shavu'ot is the time when the grain is already harvested, the fruits begin to ripen, and the grape harvest starts.
Sukkot is the festival of gathering in and storing all the produce of the fields, when the annual agricultural growth has ended, and there is time to enjoy the fruits of the past year before beginning to plant for the next year.
–Rabbi Adin Steinsaltz
From A Guide to Jewish Prayer,p.147, by Rabbi Adin Steinsaltz