Tachanun after Sukkos:
Tachanun is omitted from Davening from Erev Yom Kippur until the beginning of the month of Cheshvon.
Tzidkascha: The prayer of Tzidkascha is not recited on the Shabbos between Erev Yom Kippur and the month of Cheshvon.
 Siddur Admur; Custom brought in Magen Avraham 669:1; Shaareiy Teshuvah 131:15; Seder Hayom [in end of section regarding Sukkos]; Kneses Hagedola; Peri Chadash 131:6 [entire month of Tishreiy]; Chesed Leavraham Mayan 2:45
The Seder Hayom [from the year 1600] brings that Tachnun is omitted until the end of Tishreiy because it is a month filled with Holidays and joyous events. We have 4 different holidays and were given a time of forgiveness in this month. He therefore writes it is improper to show any sadness after having experienced all these lofty matters of the month, and rather one is to rejoice in it. [See Admur 429:8-9 regarding the month of Nissan]
Ruling of Admur in Shulchan Aruch: In 624:13 Admur rules that no Tachanun is recited between Yom Kippur and Sukkos being that it is days of rejoicing. No mention is made regarding after Sukkos.
Other Opinions: Some Poskim rule that the custom is to recite Tachanun after Sukkos. [Shaareiy Teshuvah ibid; Ashel Avraham Butchach 131]