There are certain differences in the daily prayers that apply every day of Sukkot, including the "intermediate days" (Chol Hamoed), and some that apply only on the first two full-fledged festival days (or one, if in Israel).
On the first two days, we supplement our ordinary evening, morning, and afternoon prayers with special Shabbat-like ones, replacing each ordinary amidah with one that is recited just on the three festivals.
On Chol Hamoed, we recite the ordinary prayers and ordinary amidahs, but insert the ya'aleh v'yavo passage into each amidah. (See Do I put on tefillin during Chol Hamoed?)
After the Hallel, the congregants encircle the synagogue's reading table while holding the Four KindsOn every day of Sukkot, the morning prayers are followed by the recital of the complete Hallel. The Four Kinds are an integral part of this. They (or the lulav only, according to Chabad custom) are held throughout the Hallel, and waved in all directions at various intervals. (On Shabbat, the Hallel is recited sans the Four Kinds.)
After the Hallel, the congregants encircle the synagogue's bimah (Torah reading table) while holding the Four Kinds, meanwhile offering special prayers – hoshanot – beseeching G‑d to give us an abundance of wealth in the upcoming year. (The hoshanot ceremony is not done on Shabbat at all.)
This is followed by special Torah readings (click here for our Sukkot Torah Readings Section):
On the full-fledged festival days, two Torah scrolls are taken out of the Ark. First a passage from Leviticus (ch. 22-23) is read, detailing the laws of Jewish holidays—including the mitzvot of the festival of Sukkot: dwelling in the sukkah, taking the Four Kinds, and rejoicing. This reading is divided into five aliyot (seven if it is also Shabbat), and is followed by a maftir, read from the second scroll, from the twenty-ninth chapter of Numbers, about the festival's sacrifices.
The first day's haftorah (reading from the Prophets) is from Zechariah (14:1-21), and includes a prophecy regarding the Messianic Era: that all the nations will then go to Jerusalem every Sukkot to pay homage to G‑d. The second day's haftorah, from I Kings (8:2-21), describes the dedication of the Ark of the Covenant that occurred during the holiday of Sukkot.
On the intermediate days of Chol Hamoed, the entire Torah reading concerns the sacrifices of the day. It is all read from one scroll, and divided into four aliyot. There is no maftir.
The Torah reading is followed on all the days of Sukkot by the recital of the special festival Musaf.
"May the Merciful One restore for us the fallen sukkah of David"On Shabbat of Chol Hamoed (on those years when Chol Hamoed includes a Shabbat), we recite all the ordinary Shabbat prayers, while adding the ya'aleh v'yavo into the amidahs. (The Friday Night prayers are slightly abridged.) The complete Hallel is recited, but the Four Kinds are not used and there is no hoshanot. Two scrolls are taken out of the Ark for the Torah reading: First a passage from Exodus (33:12-34:26) is read, which discusses the festivals, among other things; this reading is divided into seven aliyot. A maftir concerning the day's sacrifices is read from the second scroll, and a haftorah from Ezekiel (38:18-39:16) that discusses the war of Gog and Magog. This is followed by the festival Musaf, with occasional inserts for Shabbat.
Throughout Sukkot, when reciting the Grace After Meals, the ya'aleh v'yavo passage is added in, as is an additional one-line prayer for festivals: "May the Merciful One let us inherit the day which is all good." There is also a special line unique to Sukkot: "May the Merciful One restore for us the fallen sukkah of David [i.e. the Holy Temple]."