On Rosh HaShanah, we can say “shanah tovah umtukah,” which means “May you have a good and sweet new year.” The greeting can be shortened to “shanah tovah” (“a good year”). A 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).” Another greeting is "tizku l'shanim rabot," which means "May you merit many years."
You may hear people say “Passover, Shavuot, and Sukkot.” but strictly speaking, chag sameach is used only on the three pilgrimage festivals:
Another traditional greeting for Ashkenazi Jews on both Rosh HaShanah and Yom Kippur is a greeting, “gut yontif,” which means “Wishing you a good holiday.” Sephardi Jews may offer a greeting in Ladino, "anyada bueno, dulse, i alegre," meaning "May you have a good, sweet, and happy new year," or a Hebrew expression, "tizku leshanim rabot" meaning "May you merit many years."
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.
Whether you are greeting people in person, over the phone, online, or just signing an email, using a special phrase acknowledges the importance of theseason and expresses your special good wishes for the other person at such a time.
You can learn more about terms to use during the High Holidays in the ReformJudaism.org glossary.