Siman 600
Presented here for study, not as a practical ruling. The Shulchan Aruch rules by Sephardi custom, the Rema's glosses by Ashkenazi custom, and the Kitzur follows Ashkenazi custom. For a practical question, ask your rabbi.
א בֵּיצָה שֶׁנּוֹלְדָה בְּיוֹם טוֹב הָרִאשׁוֹן, וְכֵן מַה שֶּׁנִּצּוֹד בּוֹ, וְכֵן מַה שֶּׁנִּתְלַשׁ בּוֹ, אָסוּר בַּשֵּׁנִי. הַגָּה: וּכְבָר נִתְבָּאֵר לְעֵיל סִימָן תקי"ג (סָעִיף ה') וְסִימָן תקט"ו.
1 An egg that is laid on Rosh HaShana and the Laws of the Kiddush (of Rosh HaShana), and in it is 3 paragraphs An egg that is laid on the first day of Yom Tov, and similarly what is trapped on it (i.e. a trap set prior to R'H catches something the 1st day), and that which was detached on it (i.e. fruit from a tree detached on the 1st day etc.) is forbidden on the 2nd day (as well). Rema: And it has already been explained earlier in Siman 513 Paragraph 5 and Siman 515.
ב בְּקִדּוּשׁ לֵיל שֵׁנִי יִלְבַּשׁ בֶּגֶד חָדָשׁ (הַגָּהוֹת מַיְמוֹנִי פ' כ"ט מֵהִלְכוֹת שַׁבָּת) אוֹ מַנִּיחַ פְּרִי חָדָשׁ וְאוֹמֵר שֶׁהֶחֱיָנוּ; וְאִם אֵין מָצוּי (בֶּגֶד חָדָשׁ אוֹ) פְּרִי חָדָשׁ, עִם כָּל זֶה יֹאמַר שֶׁהֶחֱיָנוּ.
2 During Kiddush on the 2nd night (of R'H) wear a new garment) (Haagos Maimoni Chapter 29 from the Laws of Shabbos) place a new fruit, and say the blessing of Sheychianu. Yet, if there is not found (a new garment) a new fruit, nonetheless, say Sheychianu.
ג אִם חָל יוֹם רִאשׁוֹן בְּשַׁבָּת, אוֹמְרִים שֶׁהֶחֱיָנוּ, בַּשּׁוֹפָר בְּיוֹם שֵׁנִי. הַגָּה: וְיֵשׁ אוֹמְרִים לְאוֹמְרוֹ אֲפִלּוּ אִם חָל יוֹם רִאשׁוֹן בַּחֹל, וְכֵן הַמִּנְהָג בִּמְדִינוֹת אֵלּוּ (הַגָּהוֹת מַיְמוֹנִי הִלְכוֹת שׁוֹפָר).
3 If the 1st day (of R'H) falls on Shabbos, we say Sheychianu on the Shofar during the 2nd day (as the first day it is forbidden to blow). Rema: Yet, there are those that say to say (the Sheychianu blessing on the 2nd day) even if the 1st day is on a weekday. So is the custom in our countries [Haagos Maimoni, Laws of the Shofar].