It is our practice to shake our garments during the Tashlich service, while reciting the words, "Tashlich Bi’msulot Yam Kol Hatotam" ("You shall cast all their sins into the depths of the sea"). Obviously, shaking one’s garment does not automatically eliminate his sins from his record; if it did, then we would simply shake our garments each day without ever having to undergo the grueling process of repentance and self-improvement. This gesture is certainly not a substitute for repentance, but is rather a symbolic act to express our sincere desire and attempt to rid ourselves of our sins and return to the faithful service of our Creator.
One may not feed the fish in the river on Shabbat and Yom Tob; it is thus forbidden to throw food to the fish in the river during Tashlich (even inside an Erub).
It is improper for women to attend Tashlich, for a number of reasons, most importantly because of the likelihood of inappropriate mingling and socialization. If Tashlich becomes a social event, rather than an opportunity for reflection and Teshuba, then whatever is achieved is more than offset by what is lost. A person can pray for the elimination of his sins, but in the end commits even more sins through improper socialization with the opposite gender. It is therefore preferable for the women not to attend the Tashlich service. In fact, the Aruch Ha’shulhan (Rabbi Yehiel Michel Epstein, Byelorussia, 1829-1908) wrote that in places where women attend Tashlich, the men should not attend, as it is preferable not to go to Tashlich at all than to go and run the risk of improper behavior.
Summary: It is customary to shake one’s garment during Tashlich as a symbol of his attempts to rid himself of his sins. One may not feed fish in the river on Rosh Hashanah. It is preferable for women not to attend Tashlich, in order to avoid inappropriate mingling at an event that is intended as an opportunity for serious thought and introspection.
