Shulchan Arukh Orach Chayim · Siman 597

English

If it is permitted to fast on Rosh HaShana. 3 paragraphs. One eats, drinks and is merry [on Rosh Hashanah.] We do not fast, whether on Rosh Hashanah, nor on Shabbos Shuva (the Shabbos of Repentance which is between R'H and Yom Kippur). However, people should not eat to complete satiation, so that they do not reach light-headedness, and that the fear of God should remain on them.

English

There are places that have an accepted tradition that all whose routine is to fast on Rosh HaShana, and change their routine, and do not fast. They do not complete their year. Rema: One who does not fear for his soul, it is not necessary to fast all his days [Kol Bo and Minhagim] only he needs to get a Halachic cancellation like with other vows [Kol Bo].

English

One who fasts once on Rosh HaShana a Fast of Dreams (Taanis Cholom, a fast due to a certain type of dream the night before). If it is the first day (of R'H) he needs to fast the two days his whole life. If it is on the 2nd day, he fast his whole life the 2nd day alone. Rema: It is not necessary to duplicate his fast for his fast because there are those that say it is mitzvah to fast on Rosh HaShana (Terumas HaDeshen Siman 245). And he does not say the Aneinu prayer (said on other fast days) on Rosh HaShana just like we do not say on Yom Kippur (despite fasting) (Haagos Minhagim).