For whether to eat in your sukka on Shmini Atzeret, see Sukka: Shmini Atzeret.
 

Ma'ariv on Shmini Atzeret and on Simchat Torah may be said from 1 1/4 hours before sunset. Kiddush must be said after dark.

Outside Eretz Yisrael, you should eat your meals in the sukka on Shmini Atzeret, but do not make a blessing on the sukka.
 

In Eretz Yisrael, Shmini Atzeret and Simchat Torah are observed on the same day. Outside of Eretz Yisrael, Simchat Torah is the second day of what becomes a two-day festival.

We begin mentioning rain in the second paragraph of the musaf amida of Shmini Atzeret and continue until the first day of Passover.
 
The universal custom is to complete the reading of the Torah and to begin reading it again on Simchat Torah. Dancing and singing with the Torah scrolls is also a universal custom.

Saying the seasonal addition to the amida of mashiv ha'ruach… begins at musaf of Shmini Atzeret. The last time it is said is on the first day of Passover in musaf.

Here are the d'oraita Jewish festivals:

• First day of Sukkot, Passover, and Shavuot
• Seventh day of Passover
• Shmini Atzeret (8^th day of Sukkot)
• Both days of Rosh Hashana.

On these days, you:

• MUST take medicine whenever there is any question of a life-threatening disease or condition.
• MAY take medicine for a condition that affects the entire body (fever, weakness, etc.). Consult a rabbi if possible.
  Exception You may not smear substances on skin UNLESS the fever is life-threatening, in which case even smearing is permitted.
• MAY use some medicines if only part of your body is affected by a non-life-threatening disease--consult a rabbi.

Say she'hecheyanu when lighting candles for:

• Both days of Rosh Hashana,
• Yom Kippur,
• First day (if in Eretz Yisrael) or first two days of Sukkot,
• Shmini Atzeret and Simchat Torah,
• First day (if in Eretz Yisrael) or both days of Shavuot, and
• First day (if in Eretz Yisrael) or first two days of Passover.

Some people have the custom of lighting a yahrzeit candle for a deceased parent on days when yizkor is said: Yom Kippur, last day of Passover; second day of Shavuot; and on Shmini Atzeret.

Yizkor is not required to be said at all, but it is the custom in most places. Yizkor is normally said on Yom Kippur (and the final day of Passover, the second day of Shavuot and on Shmini Atzeret), when many people attend the synagogue prayer services.
Yizkor may be said anytime you wish—even when it is not a Jewish festival--and does not require a minyan.

Begin saying L'David Adonai ori v'yish'i (Psalm 27) on the first day (at night) of Elul, not the first day of Rosh Chodesh Elul (the 30th of Av). The last day to say this Psalm is 22nd of Tishrei (Shmini Atzeret). This Psalm is recited twice daily, in shacharit after shir shel yom and in ma'ariv after alenu, including on Shabbat and Jewish festivals.

Since two days of Jewish festival are observed outside of Eretz Yisrael, read the lines for the sacrifice for both days that might have been the correct day.

Tachanun is related to judgment. Tachanun is NOT said at times of din/judgment:

• At night,
• On Tish'a B'Av,
• In a house of mourning, and
• Yom Kippur.

Tachanun is also NOT said at times of simcha/happiness:

• Shabbat,
• Jewish festivals,
• Rosh Hashana, and
• Rosh Chodesh.

• Isru chag (the day after each of the Jewish festivals),
• Entire month of Nisan.

• All of Chanuka, Purim, Shushan Purim, Tu B'Shvat, Rosh Chodesh, and from Rosh Chodesh Sivan until the day after Shavuot.
• Tishrei from shacharit before Yom Kippur until after Simchat Torah (Shmini Atzeret in Eretz Yisrael). Resume saying tachanun:
  • Second day of Cheshvan, or
  • Day after isru chag of Simchat Torah (this is the more prevalent custom among Ashkenazim). Each person should follow his or her family or community custom.

• Mohel,
• Sandak, or
• Father of a boy having his circumcision.

Jewish Festivals: Medicines: When To Take
Medicine generally may not be used on the d'oraita Jewish festival days.

Jewish Festivals: Medicine for Chronic Diseases
You may take medicine on Jewish festivals (whether d'oraita or d'rabanan) for:

• Chronic diseases, such as Parkinson's disease, or
• Any disease that affects your entire body.

Jewish Festivals: Medicine for Non-Chronic Diseases
You may take medicine on Jewish festivals for non-chronic illnesses, if skipping one day will prevent cure.  You may not take medicine for non-chronic illnesses if skipping a day will just delay your being cured (unless the disease affects your entire body--in which case, you may take the medicine).

Jewish Festivals: Medicines: When To Take: D'Oraita Festival Days
Here are the d'oraita Jewish festivals:

• First day of Sukkot, Passover, and Shavuot
• Seventh day of Passover
• Shmini Atzeret (8^th day of Sukkot)
• Yom Kippur
• Both days of Rosh Hashana (even though the second day is d'rabanan).

On these days, as well as on d'rabanan Jewish festivals, you:

• MUST take medicine whenever there is any question of a life-threatening disease or condition.
• MAY take medicine for a condition that affects the entire body (illness, weakness, etc.). Consult a rabbi if possible.
  ExceptionYou may not smear substances on skin UNLESS the illness is life-threatening, in which case even smearing is permitted.
• MAY use some medicines if only part of your body is affected by a non-life-threatening disease--consult a rabbi.

Jewish Festivals: Medicines: When To Take: D'Rabanan Festival Days
You make take medicines for any reason on d'rabanan Jewish festivals--even medicines not allowed on the first day of Jewish festivals--except:

• Medicines that you smear on skin.
• If the Jewish festival falls on Shabbat (which can only be second day of Shavuot).

To fulfill the two requirements for Jewish festival evening kiddush:

1. Make Kiddush
   Say, or hear, the three Jewish festival evening kiddush blessings/segments:
   1. Borei pri ha'gafen (if on wine or grape juice—preferred option), OR
      Ha'motzi (on two challot if you have no wine or grape juice, since no chamar medina is allowed for Jewish festival evening kiddush; see Jewish Festivals: Kiddush: Challot for Kiddush) AND
   2. Asher bachar banu mi kol am.... mekadeish Yisrael v'hazmanim, AND
   3. She'hecheyanu on all Jewish festival nights, except the last two nights of Passover.  So say she'hecheyanu on:
      • Both nights of Rosh Hashana (in or outside of Eretz Yisrael),
      • First two nights of Passover (1 night in Eretz Yisrael),
      • Both nights of Shavuot (1 night in Eretz Yisrael),
      • First two nights of Sukkot (1 night in Eretz Yisrael), and
      • Nights of Shmini Atzeret and Simchat Torah (same day in Eretz Yisrael).
      Note There is never any requirement on an indvidual to drink kiddush wine (except at the Passover seder), but the kiddush wine must be drunk by one or more persons.
      Note A woman who had said she'hecheyanu when she lit the Jewish festival candles does NOT say she'hecheyanu again if she makes kiddush for herself, even when making kiddush at the Passover seder.
      Note There is no need to eat a new fruit after saying she'hecheyanu on the second night of  Shavuot, Passover, or Sukkot. This is not comparable to Rosh Hashana, since the second festival night was instituted due to uncertainty of the actual date of the holiday, while Rosh Hashana is considered to be one single, long day.
2. Establish a halachic “meal” (kovei'a se'uda).
   For how to establish a halachic meal, see Introduction to Jewish Festival “Eating a Meal” Requirements.

We are required to eat two meals on Jewish festivals, each preceded by kiddush, one in the evening and one in the morning. For both meals for each Jewish festival day (and all three meals on Shabbat), say ha'motzi over two complete loaves of bread, each of which is at least 1.3 fl. oz. in volume. 

Source of Saying Jewish Festival Kiddush
Some kiddushes are commanded by the Torah (d'oraita); the others are from Chazal (d'rabanan), as follows:
Kiddush d'Oraita

• First night of Jewish festivals
• Night of seventh day of Passover
• Night of Shmini Atzeret.

Kiddush d'Rabanan

• First day of Jewish festival
• Second night of Jewish festival
• Second day of Jewish festival
• Seventh day of Passover
• Eighth night of Passover
• Eighth day of Passover
• Shmini Atzeret day
• Night and day of Simchat Torah.

Source of Jewish Festival Kiddush Location
Saying Jewish festival kiddush at the place where you will eat your meal is a rabbinical (d'rabanan) enactment.
Jewish Festival Kiddush-Meal Quantities: Evening

• For evening kiddush for a first-night (d'oraita) Jewish festival, a minimum of 4 fl. oz. (119 ml) of wine must be blessed on and at least half must be drunk.
• For evening kiddush for a second-night (d'rabanan) Jewish festival, a minimum of 3.3 fl. oz. (99 ml) of wine must be blessed on and at least 2.0 fl. oz. must be drunk.
• For the evening meal, as on the first two Shabbat meals and for all required Jewish festival meals, a minimum of 1.9 fl. oz. (56 ml) of bread must be eaten within four minutes.

Eating Requirements for Jewish Festival Morning Kiddush
There are two separate eating requirements during the daytime. They may be combined (say/hear kiddush and start the main meal right away) but are often done separately (say/hear kiddush and then eat some light foods and beverages; the main meal is eaten later in the day).
Note Since eating and drinking requirements on all morning kiddushes (both Shabbat and Jewish festivals) are d'rabanan, the required beverage amount for morning kiddush is only 3.3 fl. oz. (99 ml) instead of the d'oraita 4 fl. oz. (119 ml) (which is required for kiddushes for Shabbat evening and all first-night Jewish festivals).

• Morning kiddush requires a halachically legal “meal” with these elements:
  • Blessing on a minimum of 3.3 fl. oz. (99 ml) of wine (or other beverage),
  • Someone's drinking at least half the beverage, followed by
  • Eating at least 1.3 fl. oz. (39 ml, or 1/6 cup) of some type of mezonot within four minutes. 
    Note If you drink at least 3.3 fl. oz. (99 ml) of wine within 30 seconds, you do not need to eat mezonot.
    Note You do not need to drink the wine or other kiddush beverage to fulfill “establishing a meal.” You may hear kiddush and then simply eat the required amount of bread or mezonot. This applies to Shabbat or Jewish festivals, evening or morning.
     

 

• The real meal (kovei'a se'uda) requires eating at least 1.9 fl. oz. (56 ml) of bread (or matza during Passover!) within four minutes. It must include enough food to satiate. 
  Note You can simultaneously fulfill the requirement to “establish a meal” and to “eat a meal” by eating one (the same) piece of bread.