I will try to discuss the Minhag of Tashlich, and try to bring out some of its many Minhagim.
There are different opinions to what Tashlich comes to symbolize.
Amongst them, the most common opinion is, it symbolizes that when Avraham took Yitschak to the "Akadah", the Satan came in the form of a river and tried to stop Avraham from doing Hashem's wishes, and when the water reached Avrahams neck he begged from Hashem ; "Hashem, you commanded me to bring my only son Yitschak, and I didn't wait a minute, right away I went, and Avraham said that if he drowns now in the water, who would bring out Hashems oneness, and Avraham cried out, "Hoshiayne Hashem Key Bahuh Mayim Ad Nafesh". Right away Hashem rebuked the Satan, and the Satan left. The "Zohar" also brings out that the "Akaidas Yitschak" took place on Rosh Hashanah, therefore we read on Rosh Hashana from the "Akaidas Yitschak ".
From the "Zohar's" reasoning above, we also do Tashlich to remember the merits of Avraham and Yitschak.
Another reason brought down by the "Yechaveh Das", that the siddur "Ohalay Yaacov" brings down a Gemarah in "Horios", that the anointing of a king is done by a spring of water, to symbolize that there rulership should spread like the flow of water, for a "Siman Tov".
And since on Rosh Hashanah we do things that bring us a Siman Tov, we also do this, and hope that Hashem will show us flowing kindness, and write us in the book of life and kindness.
As far as I have found, the earliest source for this Minhag is found in the Zohar, the earliest Halacha sefer found in is the “Maharil”.
Once we have an idea of what Tashlich symbolizes, I will like to try and clarify where Tashlich should take place.
2 WHERE TO SAY TASHLICH:
The first choice for Tashlich is by a river, with fish, out of the city.
The "Ari z"l" brings out that the main reason to do tashlich by a river is because of the "upper waters". Some say, one can also say Tashlich, by a "Be-er Mayim Chayim", the "Kaf Hachaim"brings out that since in Yerushalayim they did not have flowing water, one may say Tashlich even by a pit used to hold water, even if there presently is no water in it, and even if it is in the city.
It is brought out that the "Grah", would say Tashlich, not by flowing water.
Also, the "Chasam Sofer" would say Tashlich by a park behind his house. He would go up on a hill there, and see the "face of the river", from a distance.
There are many opinions of why it is good to have fish in the water:
The reason to say Tashlich preferably outside the city is,
Presently not found a reason for this yet
3 WHAT TIME OF THE DAY TO SAY TASHLICH:
From the "Maharil", there are those that learn out that he used to do Tashlich right after the Seudah, before Mincha.
The "Mateh Efraim" brings out, to say Tashlich after Mincah, and one can say it even after sunset till night.
The only source found to say it after Mincah, is from the “Ari z"l”, while the "Shulchan Ha-Aruch" does not say when to go.
The "Mesader Ha Levush" says to say it after Musaf, before the Seudah.
While the "Torah Lishmah" comes and says that the "Ari z"l" only said to say it after Mincha , if Tashlich is done on the first day of R"H. However, if Tashlich is done on the second day of R"H, one does it before the afternoon.
The M.B. brings out that we go to Tashlich after Mincha.
Most however bring down, that if Tashlich can't be done after Mincha, it can be done before.
4 WOMEN & TASHLICH
The "Aruch Ha Shulchan" goes out against the social gathering that sometimes happens by Tashlich, and therefore says women should not go, if they do, it is better that the men should not go there. " Rav Naftoli Hofner" brings out that because of this, it is possible it may have brought out the "Minhag Ashkenaz Haprushim", not to go to the water, but to say Tashlich in the Shul or at home.
The "Mateh Efraim" says, that because of the above, women should stop going to Tashlich.
The Sefer "Zechor L'avraham", brings out that women do have the minhag to go to Tashlich, yet as long that it doesn't bring to the mixing of men and women together.
5 SHAKING OUT THE CORNERS OF ONES GARMENT
The "Kitsur Shullchan Ha Aruch" brings out that one should shake out the corners of ones clothing, inorder to remind one to throw away ones sins and from today on, to commit ones self to find ways inorder to better his ways. Also, that his clothes should be white for now on, clear of sins.
The "Matei Efraim" brings out that one should shake out specificaly ones Tsitsis corners. The "Siddur Rav Shabtai" says to shake ones corners when he says the word " Vehsashlich".
The Kaf Hachaim says to "Davkah" shake out ones upper garments, and brings down the Zohar that says one shakes off the "Klepos" below, and hopes that above the same would be done to the Klepos deep in ones soul, thrown into the waters above.
6 PROBLEM OF BAKASHOT ON YOMTOV/SHABBOS
This is brought down by the Sefer 'Shut' "Rav Poalim", if there is a problem to say Tashlich on Yom Tov, or Shabbos; since we know that one is not supposed to ask for his needs on these days. He brings out that the "Chidah" put together, part of the Tefillos for Tashlich, amongst other Tefillos for Yom Tov that do have a request for ones needs in it. He answers this question this all true for a 'one time' personal request, yet where we have set requests in Davening, we are allowed to say them, when they were prepared for the whole congregation.
7 WHAT DO YOU SAY
Most say to at least say the last 3 Pesukim from Micha "Mi Kayl Kamocha... " and have in mind for the 13 midos of hashem. There are those that say this pasuk 3 times.
The above is the way the "Chasam Sofer" did Tashlich.
There are many other "Tachnunim" that were added on afterwards, and there are many that have the Minhag to also add to this, what the "Chidah" was Metaken.
8 WHAT IF I MISSED THE FIRST DAY
In the Siddur of "R'Yaakov May Emden", he writes to only go on the first day of R"H, yet there are those that when Shabbos falls on the first day, "Ohnes', one can do it on the second day. The "Kaf Hachaim" adds that if done on the second day, the "Ben Ish Chai" says it is better to do after Musaf.
The "Torah Shleimah" brings down that the "Ari z"l" probably said to say after Mincha only on the first day and not the second day. Because only on the first day we have the reason to say it in the 12th hour of the day, for it is "Ays Ratzon".
Most opinions hold that one can make up on the second day, there are those that say that one can make it up till Hoshanah Rabah.
The "Maharich" says that there are those that "Davkah" say it on the Thursday before Yom Kippur, since that is the day that the 13 Middos are said in the Selichos. This is also a day of mercy, uplifting, and pity. This day was the day that the First Beis Hamikdash was inaugerated in the days of Shlomo Hamelech, thus a day for prayer and repentance. Yet, it is still better to say with Tsibbur when said on the first day, and not to separate oneself from the Tsibbur.
9 WHAT IF SHABBOS FALLS ON THE FIRST DAY
There are two different opinions:
1) To say it on Shabbos and not push it off to the next day.
This opinion is learned out from the way the "Maharil" wrote about Tashlich, and the "Shvut Yaacov" adds on this, that it is so because of "Zerezem Makdimim" and the problem of carrying bread applies on Yom Tov just like on shabbos, since one is not allowed to carry "Loh Leh Tsorech" on Yom Tov.
The "Mateh Efraim" also writes to do it on the first day.
2) To push it off to the second day is found in the "Kitsur Shulchan Ha Aruch".
There are also others that bring it down, and in most places this is the Minhag,
even though allot of opinions hold to do it on the first day. I think that we are Noheg this way, like the "Mishnah Berurah", because of the Cheshash of carrying Siddurim outside the Eruv.
The "Piskeh Teshuvos" brings a Machlokes on the "Ari z"l's" opinion.
Some say even if it is in the Eruv, don't say on tfirst day, because of hidden reasons.
Others say to do it especially on the first day, with most opinions saying, that this is when it is in the Eruv, and thus no "Cheshash" of carrying.
The M.B. brings out that there are a few places that push off Tashlich when R.H. falls on Shabbos, and he thinks it is because we are worried that people would carry out of the Eruv, yet if it would be in the Eruv, than one would probably say it on the first day.
There are those that hold in the case of saying Tashlich on the second day, to say it after Musaf, since only on the first day there is reason to say it after Mincha.
10 FEEDING THE FISH
According to all opinions, it is strictly forbidden to throw bread crumbs or any other food into the water. It is also forbidden to carry the bread outside, even if there is an Eruv, and some say, even if it is not Shabbos and only Yom Tov, where carrying is generally allowed.
This is found in the "Maharil" and the "Maate Efraim"
The "Maharil" gives 2 reasons for not throwing the bread:
a) The fish are Muksah (both on Shabbos and Yom Tov) since they are lacking "being caught", one can only feed something that he can obtain, and one can only feed something that is not Muksah on Yom Tov.
b) There is a problem of taking food out of the Eruv without need, which is forbidden even on Yom Tov.