We are permitted to cook on Yom Tov in order to eat the food on Yom Tov. However, when Shabbat immediately follows Yom Tov, as this year, we are not permitted to cook on Yom Tov in preparation for Shabbat. In order to be allowed to cook for Shabbat on Yom Tov, one must designate an eruv tavshilin.
The Eruv Tavshilin is a parcel of food that is set aside before Yom Tov to be eaten on Shabbat. It should include an egg's worth of a baked item, and an olive's worth of a cooked item. These need not be baked or cooked for the purpose of the eruv, so store-bought bread and a canned cooked vegetable would suffice.
The food of the Eruv is left aside until Shabbat, at which time it may be eaten, though one need not eat it if he does not wish to.
When designating the Eruv, one holds the foods and recites a blessing, followed by a declaration:
ברוך אתה ה' א-להינו מלך העולם אשר קדשנו במצותיו וצונו על מצות עירוב
Baruch Ata Adonai Elohenu Melech Haolam Asher Kideshanu Bemitzvotav Vetzivanu Al Mitzvat Eruv.
Blessed are you God our Lord King of the world who sanctified us with His commandments and commanded us regarding the mitzva of Eruv.
בדין יהא שרי לנא לאפויי ולבשולי ולאטמוני ולאדלוקי שרגא ולתקנא למעבד כל צרכנא מיומא טבא לשבתא.
With this it will be permitted to us to bake and cook and insulate and light candles and prepare and attend to all our needs from Yom Tov to Shabbat.
Even one who does not wish to cook for Shabbat on Yom Tov should make an Eruv Tavshilin, in order to be permitted to light Shabbat candles on Yom Tov.
One roommate may designate the Eruv Tavshilin on behalf of the entire room or suite.
One should leave an all-day candle burning from before Yom Tov, in order to be able to light candles from it on the second day of Yom Tov. Another candle should be used to maintain a flame until Erev Shabbat, for Shabbat candles. On this occasion there is no option of lighting an electric shabbat candle, as it cannot be turned on during Yom Tov, so one needs to use fire.
We generally do not shower or bathe on Shabbat or Yom Tov. However, on this 3 day Yom Tov, one who feels dirty and uncomfortable may shower on Yom Tov in cold or lukewarm water, using liquid soap. Care should be taken not to wring out water from one's hair or towel. One should not use any hot water on shabbat, but may take a cold shower.
One should hear the Shofar blown on Rosh Hashana. Minimally, one should hear the first set of 30 sounds that are blown before Mussaf. If one leaves the service to attend to an urgent matter after that time, he does not need to hear makeup shofar sounds.
One should not interrupt with talk that is not part of the service from the time the blessing is recited over the shofar until the last sound is blown after Mussaf. However, if one did interrupt, he should not recite another Bracha.
Many have the custom to recite several verses by a body of water on Rosh Hashana. This ceremony is known as Tashlich. One is not permitted to throw bread into the water and feed the fish during this ceremony, despite urban legends to the contrary. Some do not have the custom to recite Tashlich at all. If one wishes to recite Tashlich and cannot do so on Rosh Hashana, it may be done any day before Hoshana Rabba.