On Rosh Hashanah, there is an idea that a person is supposed to be happy, to go home, and to eat and drink. Despite the fact that it is a day of judgment, we are excited and happy that our judgment is before our Father in Heaven, and we are confident that Hashem is going to give us a good judgment. However, it seems that many people did have the practice of fasting. So Shulchan Aruch writes that somebody who has had the practice year after year and has fasted on Rosh Hashanah at least three times may not switch from that practice and must continue to fast. If he does switch from that practice and does not fast, he will not fulfill out his year. Therefore, it is very severe if a person does do that; they should make sure to go ahead and do it. However, we discourage that practice because that way of behaving is not for everybody. In addition, anybody who does have that practice would fast only during the day, not at night. They would have to go home, make kiddush (sanctification over wine), and eat the foods; only during the course of the day would they not eat. If they are not going to eat, then they should spend the rest of the day just in Beit Knesset (synagogue), learning Torah and praying. But for the majority of people, it is appropriate to be happy, drink, and eat on Rosh Hashanah, as on every other Yom Tov (festival day).
What if a person has a bad dream? There is a concept of ta'anit chalom. So typically, if a person has a bad dream, he would go ahead and fast in order to eradicate the bad dream. In the case of Rosh Hashanah, we would discourage the person from fasting and tell him not to dwell on the dream, not to be bothered by it, and to disregard it. But if that does not help and the person feels compelled to go ahead and fast, then if it happens on the first day, they have to continue fasting. Fasting for the rest of their life on the first—if it happens on the first day, on both days of Rosh Hashanah, they would have to fast forever. If it happened on the second day, Shulchan Aruch says, then they would have to fast their whole life on the second day of Rosh Hashanah. Therefore, he says that a person should not think about it or consider it any kind of damage to them, so that they do not end up having to fast for the rest of their life.
If a person does fast on Rosh Hashanah, he does not have to make it up by fasting a different day because he did the wrong thing, since there are people who say that fasting on Rosh Hashanah is allowed. But again, the practice is discouraged. Have a great day. Shabbat Shalom.