1) On the first afternoon of Rosh Hashana, after Mincha, it is customary to recite the Tashlich prayer at a body of water, preferably one that contains living fish. (Rama 583:2)
2) If it will be too difficult to go after Mincha, one may go before Mincha. (Mateh Efraim 598:7 and Netai Gavriel Rosh Hashana 69:1)
3) Harav Ovadia Yosef zt”l (Chazon Ovadia Yomim Noraim) explains that the reason for this custom is based on the Midrash which states, “When Avraham Avinu went with Yitzchak to Mount Moriah to offer him as a sacrifice, the Satan preceded them on the way and began to persuade Avraham against this and prevent him from fulfilling Hashem’s commandment. When the Satan saw that he was unsuccessful in persuading Avraham against sacrificing Yitzchak, the Satan went before them and turned into a large river so that they would not be able to cross. Nevertheless, Avraham and Yitzchak continued on their journey and entered the water until it reached their necks. Avraham Avinu then lifted his eyes heavenwards and exclaimed: ‘Master of the World! You have appeared to me and told me, Take Yitzchak, your only son whom you love, go to the land of the Moriah, and bring him there as an offering. I have not thought twice about this and I have not hindered your command. If we drown in this river, who will sanctify your great name? Save me G-d, for the water have reached my very soul!’ Immediately, Hashem castigated the Satan and he disappeared.” The Zohar states that Akedas Yitzchakoccurred on Rosh Hashanah; it is for this reason that the Torah portion read on Rosh Hashanah deals with Akedat Yitzchak. Similarly, it is for this reason that we customarily go to river or pond on Rosh Hashanah in order to recite Tashlich on Rosh Hashanah in order to commemorate the merits of Avraham and Yitzchak.
4) It is preferable to travel to the edge of the water. Yet, according to many authorities one may recite Tashlich when one can see the river, even if he is not standing at the edge of the water.
5) The custom of Chabad is to recite Tashlich when they are so close to the water that they can see the fish. (Shulchan Menachem vol. 3 page 107)
6) If there is no river, lake, or pond in close proximity of one’s vicinity, it is permissible to recite the order of Tashlich upon a bucket of water or a running hose or faucet. (Ruling of Lubavitcher Rebbe zt”l cited in footnote of Shulchan Menachem vol. 3 page 107)
7) Women are not obligated to go to Tashlich. Yet, the common custom is for women to go. However, there were seforim (including Yalkut Yosef) that advised against women going. Therefore, a women should follow her custom.
8) One should not bring bread to Tashlich to feed the fish if he is recitingTashlich on Rosh Hashana. The reason being that one may not feed animals that he doesn’t own on Yom Tov. (Magen Avraham 583:5 as explained by the Machatzis Hashekel)
9) If one did not recite Tashlich on Rosh Hashana, it may be recited during the Ten Days of Repentance, or even until Hoshana Rabba. (See Netai Gavriel Rosh Hashana Chapter 69)