On Rosh HaShanah, we can say “Shanah tovah um’tukah,” which means “May you have a good and sweet new year.” The greeting can be shortened to “Shanah tovah” (“A good year”). The more formal expression is “L’shanah tovah tikateivu v’teichateimu”, which means “A good year, and may you be inscribed and sealed (for blessing in the Book of Life).”
You may hear people say “Chag sameach” (“Happy holiday”), but strictly speaking, chag sameach is used only on the three pilgrimage festivals: Pesach, Shavuot, and Sukkot.
Another traditional greeting for both Rosh HaShanah and Yom Kippur is a Yiddish greeting, “Gut yontif,” which means “Wishing you a good holiday.”
Special greetings on Yom Kippur include “G’mar chatima tovah,” which means, “May you be inscribed (or sealed) for good [in the Book of Life],” and “tzom kal,” which is used to wish others an “easy fast.”
You can learn more about terms to use during the High Holidays in the ReformJudaism.org glossary.