Nowadays we have a calendar and know when Rosh Ha-shana is. Thus, the first day of Rosh Ha-shana is Torah-mandated, while the second day is rabbinic. This is true for all the holidays in the diaspora as well – the first day is Torah-mandated and the second is rabbinic. Therefore, if doubts arise concerning the laws of Yom Tov or shofar, on the first day we are stringent, following the principle, “When there is uncertainty about Torah law, we are stringent.” In contrast, on the second day we are lenient, following the principle, “When there is uncertainty about rabbinic law, we are lenient.” Thus, the Sages permitted performing burials on the second day of Yom Tov and Rosh Ha-shana, for the sake of the dignity of the dead (SA 526:4; Peninei Halakha: Festivals 7:5). It is also forbidden to prepare on the first day of Rosh Ha-shana for the second day. This prohibition includes cooking, setting the table, and washing the dishes. These laws are the same as those concerning preparing on the first day of Yom Tov for the second day in the diaspora (SA 503:1; Peninei Halakha: Festivals 9:5, 2:12).
However, there are some differences between the two days of Yom Tov celebrated in the diaspora and the two days of Rosh Ha-shana. The two days in the diaspora were grounded in uncertainty as to which day the court had sanctified the month. In contrast, Rosh Ha-shana was sometimes celebrated for two days even when there was no doubt as to its date. This happened when the witnesses arrived in the late afternoon of the first day. In this case, the Sages ordained that their testimony not be accepted lest the beit din sanctify the month without leaving enough time to offer all the holiday sacrifices and to recite the special psalm for Rosh Ha-shana. However, since the first day was worthy of being sanctified, the Sages ordained that the day continue to be observed as a holiday, even once it was decided that it would not be sanctified. Since Rosh Ha-shana sometimes extended to two days due to this law, the Sages called the two days of Rosh Ha-shana “one long day” (“yoma arikhta”).
For this reason, it is unclear whether the berakha of She-heḥeyanu should be recited on the second day. In the diaspora, on the second day of Yom Tov, the berakha is recited; since the second day was established on account of uncertainty, its mitzvot are the same as those of the first day. In contrast, on Rosh Ha-shana, since the two days are considered one long day for some purposes, some say that She-heḥeyanu should be recited only on the first day. In practice, most poskim maintain that the berakha should be recited during kiddush of the second night as well as of the first, and this is the custom. However, le-khatḥila it is preferable to wear something new or to serve a new fruit at the meal on the second night of Rosh Ha-shana. Then all would agree that She-heḥeyanu is recited (SA 600:2). Clearly if a new fruit was on the table during kiddush so as to enable the recitation of She-heḥeyanu, She-heḥeyanu should not be recited again before eating the fruit.
Before blowing the shofar on the second day, Ashkenazic custom is to recite the berakha of She-heḥeyanu, while Sephardic custom is not to (SA, Rema 600:3). When possible, it is preferable for the person blowing the shofar to wear a new garment and have it in mind too when he recites She-heḥeyanu (MB ad loc. 7).