Regarding lighting of the candles on Rosh Hashana night. Let’s begin with our scenario this year, where the Holiday is on Friday night which coincides with Shabbat. So therefore, the ladies would light candles like they light very Friday night eighteen minutes before sunset, and they would make a Beracha, ‘Lihadlik Ner Shel Shabbat Viyom Tov.’ They would add Yom Tov because of the Holiday. That would be, 18 minutes before sunset.
Regarding, Saturday night, which is the second night of Yom Tov, ladies are also required to light candles. The question is, when do they light candles on the second night of Yom Tov? Halacha says, it’s most proper to light them when the men come home from Shul, right before you sit down to have the meal. At that time, the ladies should go to the candles and light them. They make the Beracha on the second night, ‘Lihadlik Ner Shel Yom Tov’, as it’s obviously no longer Shabbat.
One has to keep in mind, that on Saturday night, you can only light candles from an existing flame. So therefore, one has to prepare an existing flame, a candle that is there which was prepared in order to take the flames. But there is a problem, that many people are not aware of. After the lady lights the candles from the existing flame, sometimes just without thinking she blows it out. Or she shakes it out. According to the Halacha, it’s definitely forbidden to extinguish candles on Yom Tov. While it’s permissible to light from an existing flame, but to extinguish is for sure forbidden. So you have to tell them, to very carefully just place it down and to let it go out by itself. But not to shake it, and certainly not to blow it out on the holiday.
There is no Shehechiynau made at the time of Hadlakat Nerot. The ladies rely on the Shehechiynau that is made on the Kiddush.
Now once already we are on the subject of Nerot, we have a custom that‘s brought down in Halacha, that since this Friday night is also Yom Tov, so we shouldn’t say ‘Bame Madlikin’. We shouldn’t say Bame Madlikin even though normally every Friday night we say ‘Bame Madlikin’, which discusses how to light candles, and what oils are kosher, and what wicks are kosher, and which ones are not kosher. It’s our custom to read it every Friday night. However, we skip it when a holiday comes out on a Friday night. Different reasons were given. One reason is because, not all the laws are the same on Shabbat as Yom Tov regarding kosher wicks and oils. So therefore, since the laws vary from Shabbat to Yom Tov, so we skip it. Other reasons are also brought down in the Mishna Berura in Seman 270.
Lastly, going back. regarding making the Beracha on the Nerot Shabbat. There is a Machloket (argument) as to which comes first. The lighting or the blessing? Some make the Beracha first, which would make logical sense. The logic being always to make the Beracha and then perform the Mitzvah. Others light first and then they make the Beracha. The logic there is because they hold that making the Beracha brings on acceptance of Shabbat, and how could you light the candles if it’s Shabbat already? But that logic would apply for Shabbat, but on Yom Tov everybody would agree, you should make the Beracha first and then light. It’s because you are allowed to light on Yom Tov. So therefore, the Beracha is not an acceptance of anything. And therefore, it’s proper to make the Beracha ‘Lihadlik Shel Yom Tov’ first, and then to light. This applies even to those who light first the rest of the year. That’s for Shabbat. However, for Yom Tov, it’s proper to make the Beracha first and then light.