When the first day of Rosh Hashanah falls on Shabbat, a number of passages are added to the prayer service that mention Shabbat. One who forgets to add these insertions, and does not mention Shabbat in the Amida on Shabbat Rosh Hashanah, must repeat the Amida.
The Shulhan Aruch (Orah Haim 598) writes that although generally "Sidkatecha" is omitted from the Minha service on Shabbat if it is a festive occasion, such as Yom Tob, nevertheless, "Sidkatecha" is recited during Minha on Shabbat Rosh Hashanah. "Sidkatecha" includes the expression, "Mishpatecha Tehom Rabba," which speaks of G-d’s judgment, and it is therefore especially appropriate for Rosh Hashanah. Hence, it is recited despite Rosh Hashanah also being a Yom Tob. Some Ashkenazim omit "Sidkatecha" from Minha on Shabbat Rosh Hashanah, as noted by the Rama (Rav Moshe Isserles, Cracow, 1530-1572), but the standard practice among Sephardim is to recite "Sidkatecha" in such a case.
On Mosa’eh Shabbat, the second night of Rosh Hashanah, we add the section of "Va’todi’enu" to our Amida prayer at Arbit. This recitation takes the place of "Ata Honantanu" which is normally added to the Arbit prayer on Mosa’eh Shabbat. On a normal Mosa’eh Shabbat, if one forgot to recite "Ata Honantanu" in Arbit, and then ate before reciting Habdala, he must repeat the Amida. The Sages enacted this law as a "penalty" of sorts for the person who both forgot to add "Ata Honantanu" and also made the mistake of eating before Habdala. However, Hacham Bension Abba Shaul (Jerusalem, 1924-1998) writes in Or Le’sion (vol. 3) that this unique law does not apply to "Va’todie’nu." Thus, one who forgets to add "Va’todi’enu" to the Amida when Mosa’eh Shabbat is Yom Tob does not repeat the Amida, even if he mistakenly eats before reciting the combination of Kiddush and Habdala. Hacham Bension explains that the recitation of "Ata Honantanu" was enacted by the Ansheh Kenesset Ha’gedola ("Men of the Great Assembly," the group of leading Rabbis at the beginning of the Second Commonwealth), who also imposed a penalty. The text of "Va’todi’enu" was instituted much later, during the times of the Amoraim, and they did not impose such a penalty.
As we know, it is customary on the first day of Rosh Hashanah to observe the practice of Tashlich, going to a river and reciting a number of verses related to G-d’s compassion and forgiveness, including the verse, "Ve’tashlich Be’msolot Yam Kol Hatotam" ("You shall cast all their sins to the depths of the sea" – Micha 7:19). Although the Shulhan Aruch does not mention this practice, it is mentioned by the Rama, in his glosses to the Shulhan Aruch, and it was taught also by the Arizal (Rav Yishak Luria, 1534-1572), as mentioned in Sha’ar Ha’kavanot. The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in his work Od Yosef Hai, decries the fact that some people view Tashlich as some magical way of atoning for their sins. Tashlich is symbolic of the casting of our sins into the "depths of the sea," but this requires the long, intensive process of Elul and the High Holiday period during which we pray, repent and make a sincere commitment to change.
The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) observes the custom of some communities to delay Tashlich to the second day of Rosh Hashanah when the first day falls on Shabbat. This is done to protect against possible violations of Shabbat by people who will want to bring their Mahzor with them and might bring it outside the Erub. However, all leading Sephardic Poskim, including the Ben Ish Hai, Hacham Bension, and Hacham Ovadia Yosef, ruled that Tashlich should be performed on the first day of Rosh Hashanah even if it falls on Shabbat – and, in fact, especially if it falls on Shabbat. The Arizal taught that Tashlich should be recited close to sunset on the first day of Rosh Hashanah, because it was then – at the very end of the day on Rosh Hashanah – when Adam was forgiven for his sin of eating from the forbidden tree. This time is the most auspicious time for praying for forgiveness, because this is the time when Adam earned forgiveness. Therefore, Tashlich is especially appropriate on Shabbat afternoon, which is an "Et Rason" – a time when we have a unique opportunity to find favor in G-d’s eyes. Therefore, the custom among Sephardim is to perform Tashlich on the first day of Rosh Hashanah even if it falls on Shabbat. As for the concern that someone might bring his Mahzor to Tashlich, Hacham Bension responds that this concern arises only with regard to bona fide Misvot such as Shofar, as a person might be so anxious about performing the Misva that he will forget about the Shabbat restrictions. For this reason, the Sages suspended the Misva of Shofar when Rosh Hashanah falls on Shabbat. When it comes to Tashlich, however, which is just a custom, there is no such concern, and so Tashlich may be performed on Shabbat.
If one did not, for whatever reason, recite Tashlich on the first day of Rosh Hashanah, he recites it on the second day, in which case he should recite it immediately after Mussaf. When reciting Tashlich on the second day of Rosh Hashanah, one should specifically not recite it late in the afternoon, as this is a time of judgment.
Generally, it is proper not to eat a meal on the afternoon before Yom Tob, so that one begins Yom Tob with a hearty appetite. Nevertheless, when the first day of Yom Tob is Shabbat, it is permissible to eat Se’uda Shelishit in the afternoon, even though that night is Yom Tob. As the Mishna Berura explains, since eating Se’uda Shelishit fulfills a Misva, it is allowed. If possible, one should recite Minha Gedola earlier in the afternoon, and then have an early Se’uda Shelishit. (However, even if one recites Minha early, he must remember not to recite Tashlich until later in the day, before sundown, as discussed.)
Women light candles on the second night of Rosh Hashanah at the conclusion of Shabbat. Before lighting, they must recite, "Baruch Ha’mabdil Ben Kodesh Le’kodesh."
The Kiddush on the second night of Rosh Hashanah which falls on Mosa’eh Shabbat is a combined Kiddush and Habdala, following the sequence known by the acrostic "Yaknehaz" ("Yayin," "Kiddush," "Ner," "Habdala," "Zeman"). Usually, on Mosa’eh Shabbat, when we recite the Beracha over a candle, we should use an Abuka ("torch"), meaning, at least two wicks that are combined. When Mosa’eh Shabbat is Yom Tob, however, this poses a problem, because the candle may not be extinguished. It is therefore advisable to purchase before Yom Tob specially-prepared, small "Abukot" for this purpose. If one does not have such a candle available, he should simply recite the Beracha over his Yom Tob candles. Hacham Bension writes that one should not combine two candles to form an Abuka, because it would then be forbidden to separate them until the fire is extinguished.
Summary: When the first day of Rosh Hashanah falls on Shabbat, according to Sephardic practice, "Sidkatecha" is recited at Minha, and Tashlich is recited late in the afternoon, as usual. Se’uda Shelishit may be eaten during the afternoon, though it should preferably be eaten earlier in the afternoon, if possible. "Va’todi’enu" is added to the Arbit prayer on the second night of Yom Tob. Women recite "Baruch Ha’mabdil Ben Kodesh Le’kodesh" before lighting candles on the second night of Yom Tob. Kiddush and Habdala are combined on the second night, including the Beracha over a candle. Ideally, one should obtain before Yom Tob a small candle with two or more wicks for this purpose. If no such candle is available, he recites the Beracha over the Shabbat candles.