Shehechiyanu: The blessing of Shehechiyanu is recited in Kiddush on both the first and second day of Rosh Hashanah. [One who has not said Shehechiyanu by candle lighting, as is the case by men, is to say Shehechiyanu by Kiddush.] See below regarding having a new fruit on the second night by Shehechiyanu.
Yaknaha”z: When the second day of Rosh Hashanah falls on Motzei Shabbos one says Yaknaha”z in the order of Kiddush of the second night. The blessing of Besamim is omitted from the order of Yaknaha”z. The custom is to say the blessing of Meorei Haeish over the Yom Tov candles that were lit. We do not place the candles together, and nor do we place our nails towards them. Rather we simply look at the candles after the blessing and then continue with Yaknaha”z. See chapter 1 Halacha 26 for the full details of this matter!
Rosh Hashanah that coincides with Shabbos: When [the first night of] Rosh Hashanah falls on Friday evening, the following passages prior to Kiddush are read in an undertone: shalom aleichim, eishes chayil, mizmor ledavid Hashem ro’i, da hi se’udasa.
The day Kiddush: By the day Kiddush of Rosh Hashanah one says the verse Tiku and Ki Chok etc. One does not say Eileh Moadei or Askinu. When Rosh Hashanah falls on Shabbos, the following passages are said prior to the verse of Tiku. They are to be said in an undertone: Mizmor Ledavid; Askinu; Veshamru; Im Tashiv; Da Hi; Zechor; Al Kein.
The new fruit-Taking a new fruit by Kiddush of the second night:
By Kiddush of the second night of Rosh Hashanah it is accustomed to place a new fruit in front of the person making Kiddush, or for him to wear a new garment, and intend to exempt it with the blessing of Shehechiyanu. Upon him saying the blessing of Shehechiyanu he is to look at the fruit. [Immediately after Kiddush, after drinking the wine but before washing for bread, he is to say the blessing of Haeitz and eat the new fruit. The person who said Kiddush is not to make any interval, even of speech, between Kiddush and eating the new fruit. The Chabad custom is to eat a Kezayis of the new fruit. [See Q&A] One may eat other fruits to accumulate the total of a Kezayis. If he ate a Kezayis he is to recite an after blessing afterwards, prior to washing for the meal. If he did not eat a Kezayis of fruit then an after blessing is not said. The custom is that also the people who listened to the Kiddush eat the new fruit, saying a blessing of Shehechiyanu and then Haeitz over it. See Q&A!]
If one does not have a new fruit or garment: If one does not have a new fruit or new garment available, he is nevertheless to say the blessing of Shehechiyanu in Kiddush.
Q&A on the new fruit
How much of the new fruit is one to eat and is an after blessing said?
The Chabad custom is to eat a Kezayis of fruit and say an after blessing. Other Poskim however write that there is no measured amount of fruit that one is required to eat. One may choose to eat less than a Kezayis, in which case an after blessing is not recited, or one may choose to eat a Kezayis in which case an after blessing is recited before the meal. If one forgot to recite an after blessing before the meal then he is to recite an after blessing during the meal as soon as he remembers.
Are also those who listened to Kiddush to eat the new fruit after Kiddush?
The custom is that also the people who listened to the Kiddush eat the new fruit, saying a blessing of Haeitz and Shehechiyanu [see next] over it. One first says the blessing of Shehechiyanu and only afterwards recites the blessing of Haeitz.
Do the listeners of the Kiddush say Shehechiyanu prior to eating the new fruit?
Those who listened to the Kiddush that was said by the leader of the household do not need to say the blessing of Shehechiyanu prior to eating the new fruit, [unless they did not have it in mind or they made an interval, such as speaking of unrelated topics in between]. Nevertheless, even if no interval was made, the custom is to say their own blessing of Shehechiyanu over the fruit.
If the leader of the home has a fruit which for him is not new but for a household member is new, who should say Kiddush?
Some Poskim rule that the leader of the home may say Kiddush as usual, until the blessing of Shehechiyanu, which is then said aloud by the household member for which to him the fruit is still new.
May one use the Rimon as the new fruit for the second night?
The Rimon is not to be used as the new fruit being that one is required to eat it on the 1st night as part of the Simanim. Nevertheless, there are those which use it as the new fruit for the 2nd night.
 See 271/19; 473/8
 Sefer Haminhagim [Hebrew] p. 39; English p. 55; Hagada of Rebbe p. 8
 600/5-6; Siddur Admur; Tur 600; Michaber 600/2-3; See Chapter 1 Halacha 24 for the background of this matter.
 If however he said the blessing by candle lighting he does not repeat the blessing by Kiddush. [Mateh Efraim 619/12; Sefer Haminhagim p. 128 [English]
 599/1; 491/4 “When [the night of the Seder] falls on Motzei Shabbos one needs to say Havdala after Kiddush, and if he has a candle he is to say a blessing over it after the blessing of “Kiddush” but prior to the blessing of Havdala. After Havdala one is to say the blessing of Shehechiyanu”
 Sefer Haminhagim p. 118 [English]
 Sefer Haminhagim p. 120 [English]; See Otzer Minhagei Chabad p. 139-140
Other opinions: Some write that according to Admur in the Siddur which does not differentiate between R”H and other Holidays one is to recite Eileh Moadeiy also prior to the Kiddush of the day of R”H just like by any other Yom Tov. [Shaar Hakolel 40/18; Kuntrus Hasiddur p. 23; See Mateh Efraim 597/3]
 600/6; Siddur; See Chapter 1 Halacha 24 for the background of this matter.
 In 600/6 Admur writes it is proper to place a new fruit. In the Siddur he writes it is the custom to place a new fruit. Piskeiy Hassidur  explains that Admur in the Siddur is following the opinion of the Shaar Hakavanos p. 90 and Mishnas Chassidim which rules there is no need to have a new fruit placed. He hence writes that it is merely the custom to do so and not that one should do so.
 See Q&A that one is not to use the Rimon as the new fruit being that one is required to eat it on the 1st night as part of the Simanim.
Using grape juice that requires Shehechiyanu: If one has grape juice that requires Shehechiyanu nevertheless it should not be used to make Kiddush unless one does not has wine available, as initially it is a Mitzvah Min Hamuvchar to recite Kiddush over wine. [ibid; 272/2; M”A 600/1]
 So is the wording in 600/6. However in the Siddur Admur writes “it is customary to place a new fruit in front of oneself when saying Kiddush” and he makes no mention of wearing a new garment.
 Siddur; second option in 600/6; M”A 60/1; Mateh Moshe 823; M”B 600/4; M”E 600/8
Must he hold the fruit in his hand when saying the blessing of Shehechiyanu? No. [Poskim ibid] In Admur 600/6 regarding the Shehechiyanu wine he gives two options: a) to hold the cup of grape juice in his hand while saying the blessing; b) To look at the grape juice while saying the blessing. So rules also M”B 600/4; In the Siddur Admur simply mentions to look at the fruit.
 Sefer Haminhagim p. 118 [English]; Sefer Hasichos 1945 p. 4 [“so was custom in Lubavitch”]; Machzor Chabad; Mateh Efraim 600/6; Ashel Avraham Butchach 600; Darkei Chaim Veshalom 707; Machatzis Hashekel 690/1 [regarding not making an interval] See Otzer Minhagei Chabad 147-149 that in 1948 the Rebbe Rayatz questioned whether one is to eat it before or after washing and in conclusion he ate the fruit immediately after Kiddush.
Why is the drinking of the wine not considered an interval between the Shehechiyanu and eating of the fruit? See Ashel Avraham ibid
 Piskeiy Teshuvos 600/1; See Ashel Avraham ibid; Machatzis Hashekel ibid
 27 grams [See Shiureiy Torah p. 193]
 Sefer Hasichos 1945 p. 4; Otzer 148
 M”E 600/10; Otzer Minhagei Chabad p. 82; See Admur 174/6; Seder 4/13; Ketzos Hashulchan 31/3; See SSH”K 47 footnote 222 regarding having in mind to eat the fruit also during the meal in order to avoid an after blessing. See also Halacha 6 regarding this matter on the first night when eating the apple.
 M”E ibid
Avoiding eating a whole fruit: One is to avoid eating a whole fruit that is less than a Kezayis as it is questionable as to whether it requires an after blessing. [Seder 8/9; M”E ibid] Thus those that are eating Rimon as a new fruit are to avoid eating less than a Kezayis of whole Rimon seeds as this enters into doubt regarding the after blessing. [Piskeiy Teshuvos 600/1]
 600/6; See Chapter 1 Halacha 24 for the full background of this matter.
 Sefer Hasichos 1945 p. 5; Otzer Minhagei Chabad 147-149 that so was custom of Rebbe Rayatz; Hiskashrus 898 writes that in Lubavitch, and so was the custom of the Rabbeim, they would intentionally eat a Kezayis and then say an after blessing. The Rebbe would eat two dates and then recite an after blessing prior to washing. [Otzer 149]
 Mateh Efraim 600/10
 M”E ibid
 174/6; Seder 4/12; Ketzos Hashulchan 31/1-3
 Mateh Efraim 600/10
 The reason: Although from the letter of the law there is no need for any of the listeners to eat from the new fruit being that its entire purpose was to remove the doubtful blessing of Shehechiyanu that was said by the person saying Kiddush, nevertheless the custom is for everyone to eat it as a good omen for the new year. [M”E ibid]
 Seder 11/12 based Halachos Ketanos
Other Poskim: Some Poskim rule one is to first say Haeitz and then Shehechiyanu and so is the custom of many. [Ashel Avraham Butchach 600; See Piskeiy Teshuvos 225/12 for different customs and Poskim regarding this matter.]
 Piskeiy Teshuvos 600 footnote 4
 See Ashel Avraham Butchach 600
 See Machatzis Hashekel 600/1; Ashel Avraham Butchach 600 “They may say the blessing upon eating it even though they heard Shehechiyanu while seeing the fruit by Kiddush”.
 Implication of M”E ibid that states the listeners are to say Shehechiyanu and does not mention any differentiation.
 Mahril Diskin Kuntrus Achron 81; Mikraeiy Kodesh 4; Piskeiy Teshuvos 600/2
 The person saying Kiddush then answers Amen hence being Yotzei the blessing of Shehechiyanu over the holiday. It however does not help for the person saying Kiddush to say Shehechiyanu and have in mind to exempt the household member with the blessing over the fruit as some Poskim rule that the blessing of Shehechiyanu, like all blessings of praise, can only be said for another if the person saying it is also obligated in the blessing. [See P”M 225 A”A 1; Ashel Avraham Butchach 600]
 See Halacha 7 and Q&A there!
 Moed Lekol Chaiy 12/15; Piskeiy Teshuvos 600/1 [Accordingly they do not eat Rimonim on the 1st night as part of the Simanim.]