Rosh Hashana is the Jewish New Year which is both a holiday as well as a day of Judgement. Following Rosh Hashana until Yom Kippur, there is a period called Aseret Yemei Teshuva, the Ten Days of Repentance. The laws and customs for Rosh Hashana and Aseret Yemei Teshuva are outlined below:
- The beracha on candle lighting for Rosh Hashana is lihadlik ner shol yom tov, without mention of Yom Hazikaron. 
- It is a mitzvah from the Torah to hear the Shofar blows on Rosh Hashana. 
- The proper beracha is lishmoa kol shofar. However, if one recited litkoa shofar or al tekiat shofar, he fulfills his obligation. 
- The sephardic custom is that on the first day of Rosh Hashana, the beracha of shehecheyanu is recited after lishmoa kol shofar.  If the first day is Shabbat and therefore we do not blow the shofar, shehecheyanu is recited on the second day after lishmoa kol shofar.  If a sephardic person is blowing shofar for an ashkenaz congregation on the 2nd day of Rosh Hashana, then one of the ashkenaz congregants should recite the shehecheyanu. If the sephardic shofar blower, recites shehecheyanu anyway, the congregation has nevertheless fulfilled their obligation of reciting it. 
- The congregation should stand while the Baal Tokeah makes the Brachot on the shofar and then they may sit for the blows prior to Mussaf. For the blows during Mussaf and the Chazarat HaShatz, the congregation should stand.
- One who couldn't go hear the shofar with a congregation and is having it blown for him at home, must stand for the shofar blowing. 
- The custom is to stand even for the tekiot which are referred to as the tekiot dimiyushav, the 1st 30 tekiot, 3 sets of Tashrat, Tashat and Tarat.  A weak, ill, or old person can be lenient. 
- The 30 tekiot after that are referred to as the tekiot dimeumad because one is required to stand for those. However, if one sat during these tekiot he has nevertheless fulfilled his obligation. 
- One should refrain from talking from when the beracha is recited until after all 100 tekiot are blown.  One should also try not to cough or yawn during the shofar blowing so as not to confuse any listeners. 
- If one is unsure if he heard the shofar or not, he should hear it again without a beracha. 
- Ideally, the shofar blower should try to blow it from the right side of his mouth, but this certainly doesn't prevent one from fulfilling his obligation if he doesn't. 
- There is a minhag of dipping the challah in honey on Rosh Hashana. Some only dip the challah in honey, while others dip the challah in salt as well as honey. 
- There is a minhag to eat certain fruits and vegetables as a good omen for the coming year on both nights of Rosh Hashana. Some have this minhag only the first night. 
- If one is eating dates and other fruits, one should make a HaEitz on the dates because they’re from the seven species with which Eretz Yisrael is praised. Some say that one doesn’t make a HaAdama on vegetables during the meal. 
- Some say that one should say the Bracha of HaEitz, eat a little of the fruit, then make the Yehi Ratzon before continuing to eat, while others hold that one should say the Yehi Ratzon prior to the Bracha. 
- One may say the Yehi Ratzon’s with Hashem’s name.
- These fruits and vegetables include: dates, promegrantes, beets, gourd, black eyed peas, fish head, and apple. 
- One should eat these fruits and vegetables in the meal after having eaten a Kezayit of bread. 
Sleeping on Rosh Hashana
- It's a proper minhag not to sleep on Rosh Hashana. Someone who has a headache should sleep a little after midday in order to perform the mitzvot and prayers better. 
- After the meal a person should go to the shul to learn according to his ability. Some have the practice of reading Tehillim twice over Rosh Hashana. A person who usually studies halachot should learn halachot rather than say Tehillim. In any event, one shouldn't waste time on this holy day of Rosh Hashana and wasting one's time on Rosh Hashana is like sleeping.
Minhag of Tashlich
- The minhag is to say Tashlich near an ocean or river on the first day of Rosh Hashana.  The minhag is to say it after mincha 
- The custom is to recite certain pesukim that are printed in the books. 
- In the event that the first day of Rosh Hashana falls out on Shabbat there is a dispute among the poskim if one should still recite tashlich that day. 
- It is preferable for there to be living fish in the body of water. 
- Many have the custom of shaking out their pockets as a symbol of throwing at one's sins. 
- Many poskim suggest that men and women should avoid being there at the same time so that it doesn't become an inappropriate, mixed social scene on our day of Judgment. 
- If one didn't get a chance to do it on the first day, one should still do it on the second day after mussaf. 
- When the first day of rosh hashana falls out on Shabbat, some poskim say to postpone it, while others say just to say tashlich then. 
Fasting on Rosh Hashana
- It's forbidden to fast on Rosh Hashana.  Nonetheless, it's permissible to allow prayers to extend past chatzot (halachic midday) and fast until the afternoon. 
Prayers of Rosh Hashanah
- If one forgot to switch from Atta Kadosh to HaMelech HaKadosh the first night of Rosh Hashana one doesn't have to repeat Shemoneh Esrei as long as one said the Yom Tov Shemoneh Esrei, however, if one said the weekday Shemoneh Esrei or on the day of Rosh Hashana or the second night of Rosh Hashana one should repeat Shemoneh Esrei. 
- Some poskim encourage crying during the prayers of Rosh Hashana while some forbid it. 
- If one forgot Yaaleh VeYavo in Benching on Rosh Hashana during the day one does not have to repeat benching, but during the nighttime one must repeat benching. 
- Chazon Ovadia Yamim Noraim pg. 62. see there where he writes that it wouldn't be a problem of a hefsek between the beracha and the candle lighting to mention yom hazikaron, but lechatchila one shouldn't do say it.
- Yalkut Yosef 583:1 (Moadim pg. 36) from the pasuk in Bamidbar 29:1 יוֹם תְּרוּעָה, יִהְיֶה לָכֶם
- Yalkut Yosef Moadim pg. 36
- Yalkut Yosef Moadim pg. 36, Chazon Ovadia Yamim Noraim pg. 116
- Yalkut Yosef Moadim pg. 36,48, Chazon Ovadia Yamim Noriam pg. 112, 116
- Yalkut Yosef pg. 36, Sh"t Yabea Omer 1:29:11
- Yalkut Yosef 583:3 (Moadim pg. 36)
- Sh"t Haridbaz 4:25
- Mishna Berura 585:2.
- Shaar Hatziyun 585:2.
- Mishna Berura 592:2
- Chayei Adam 141:9. Sh"t Minchat Yitzchak 3:44 and 4:47 says that Asher Yatzar may be recited during this time.
- Yalkut Yosef Moadim pg. 40
- Yalkut Yosef Moadim pg. 37, Chazon Ovadia Yamim Noraim pg. 161
- Yalkut Yosef Moadim pg. 41
- Magen Avraham 583:1, Mishna Brurah 583:3, Kitzur S"A 129:9
- Nitai Gavriel (Rosh Hashana p. 209, n. 17) writes that the Chazon Ish and Stiepler's practice was to eat challah without salt on Rosh Hashana. See there for other sources on this matter.
- Kaf HaChaim 583:4. See Nitai Gavriel (Rosh Hashana p. 209) for different minhagim about whether to dip the challah in salt on the same side as it is dipped in honey or on the other side. See there also for a difference in minhagim whether to dip the challah in salt before dipping it in honey or afterwards.
- The Gemara (Horiyot 12a) says that a person should see gourds, fenugreek, leek, beets, and dates (though these definitions are the subject of controversy) on Rosh Hashana as a good omen. The Gemara (Keritut 6a) records the same statement with the text that a person should eat these fruits and vegetables as a good omen. Beit Yosef 583:1 notes the different versions and rules in S”A 583:1 that a person should eat these foods as a good omen. Kaf HaChaim 583:6 writes that if one can’t eat a certain food, he may just look at it and say the Yehi Ratzon nonetheless. Nitei Gavriel 29:24 agrees.
- Eliyah Rabba 583:1 writes that the minhag is to eat simanim on both nights of Rosh Hashana. Machazik Bracha 583:2, Chazon Ovadyah (p. 93), Teshuvot VeHanhagot 2:266 Yalkut Yosef Moadim pg. 29, Halichot Olam 2:[g. 227 and Rivevot Efraim 6:308:1 agree. However, Bnei Yisaschar 2:11 and Eishel Avraham MeButchach 583 explain the minhag of eating the simanim only on the first night of Rosh Hashana.
- Chazon Ovadyah (p. 98) writes that one shouldn’t make a Bracha upon the cooked vegetables because they’re considered as part of the meal. Halichot Shlomo (1:18), however, writes that one should make a HaAdama upon the gourd and exempt the other vegetables. To avoid all doubt, Nitai Gavriel 29:18 writes that one should make a HaAdama on a banana. Chut Shani (Rosh Hashana p. 48) agrees. It is noteworthy to mention that S”A 211:2 rules that when eating a food which is HaEitz and a food which is HaAdama one should make the bracha on the food which one prefers first.
- Halichot Shlomo (1:17) writes that one should make the HaEitz upon the dates because they’re from the seven species with which Eretz Yisrael was praised and they come before pomegranates on that list (see S”A 211:4). Ben Ish Chai (Nitzavim #4) and Nitei Gavriel 29:4 agree. Kaf HaChaim 583:13 records the practice of some people to make a HaEitz on the apple to exempt the dates. He explains that perhaps they brought out the apple first, and in such a case, one need not wait for the dates to come out to make the bracha upon them. Rivevot Efraim 8:558:3 writes that after making HaEitz on the dates, one may eat the apples prior to the pomegranates because the rules of precedence apply only regarding the Bracha, not regarding when each food should be eaten.
- Magen Avraham 583:2 quotes the Magalei Tzedek as saying that one should say the Yehi Ratzon in between the Bracha and eating. He explains that the Yehi Ratzon is a prerequisite for eating and thus does not constitute an interruption. The Magen Avraham, however, argues that it’s not such a requirement to say the Yehi Ratzon, and as such it is preferable to say the Yehi Ratzon after taking a bite. Mishna Brurah 583:4 and Chazon Ovadyah (p. 93) agree with the Magen Avraham.
- Rav Hershel Schachter (quoted in Halachipedia Article 5773 #3) said that it makes sense to say the Yehi Ratzon before the Bracha so that the Yehi Ratzon can function as a Tefilla and the eating as an action enhancing the Tefilla. Teshuvot VeHanhagot 2:266 agrees. See Chazon Ovadyah (ibid.) who quotes the Chemdat Yamim who argues that one shouldn’t say the Yehi Ratzon prior to the Bracha because that is a violation of asking for one’s personal needs before praising Hashem (See Gemara Brachot 32a).
- Mishna Brurah 583:2 writes that the text of the Yehi Ratzon begin with Yehi Ratzon Milfanecha Hashem Elokeinu VeiyLokei Avoneitu with Hashem’s actual name. Teshuvot VeHanhagot 2:266 and Chazon Ovadyah (p. 93) agree. Nitai Gavriel 29:22, however, writes that from some poskim it seems that there was a text without Hashem’s name. Rav Shlomo Zalman’s minhag (Halichot Shlomo p. 12 note 70) was to say the first Yehi Ratzon with Hashem’s name and the others with Avinu SheBaShamayim. He explained that he did so because it’s difficult to have the proper kavana when saying Hashem’s name. Similarly, the Stiepler’s minhag (Orchot Rabbenu vol 2, p. 175) was to say them without Hashem’s name.
- Torat HaMoadim 4:1
- Chazon Ovadyah (yamim Noraim p. 93)
- Chazon Ovadyah (Yamim Noraim, pg 183-4), Kaf HaChaim 584:38,39, Ben Ish Chai Parashat Nitzavim:11, Yalkut Yosef Moadim page 33
- Yalkut Yosef (Moadim p. 33), Sh"t Yechave Daat 3:44
- Chazon Ovadyah (Yamim Noraim, pg 186), Yalkut Yosef Moadim page 44. The name of this ritual comes from a verse in Michah 7:19 which says וְתַשְׁלִיךְ בִּמְצֻלוֹת יָם, כָּל חַטֹּאותָם. This minhag is mentioned by the Rama 583:2. Mishna Berura 583:8 says that this is in remembrance of the akedat yitzchak, because the midrash says that on his way, Avraham had to cross a river, and when the water level was up to his neck he called out to Hashem and Hashem saved him. Rama in Torat HaOlah 3:56 explains that when one goes to an ocean or rivers and sees the greatness of Hashem's creations, he will immediately regret his sins and will be forgiven. The Kaf Hachayim 583:30 calls this an Ashkenazic minhag but adds that the custom in his times in Jerusalem was to say it. Maaseh Rav 209 writes that the minhag of the gra and his student Rav Chaim Volozhin was not to do tashlich at all, but Rav Moshe Harari in Mikraei Kodesh Rosh Hashana 14:note 4)writes that many people who generally follow the customs of the Gra do not follow him for this.
- Chazon Ovadia Yamim Noraim page 186. Rav Moshe Shternbuch in Teshuvot Vihanhagot 1:346 agrees but says that mincha should be said earlier in the day so tashlich could be said immediately afterwards because he doesn't think it is so simple that it's permissible to carry the siddur. Also see Moadim U'Zmanim 1:34 where he discusses this issue.
- Yalkut Yosef Moadim page 35
- Yalkut Yosef Moadim page 35, Sh"t Yabea Omer 4:47, Sh"t Yechave Daat 1:53, Halichot Olam 2:pg. 231 say that you should do it on that day but if there is no eruv then you cannot carry a book with you. If there is a concern that people will carry books anyway, it should be done the next day. If there is an eruv however, one should do it. If one normally doesn't rely on the eruv to carry, in this case he can give it to a child to carry for him.
- Mishna Berura 583:8. Mishna Berura elaborates that this is symbolic of our wish to be immune to the ayin hara and that we want to be as fruitful as the fish of the sea. Mateh Ephraim 598:4-5 also says it should be done with fish, but criticizes the minhag that some have of feeding the fish crumbs of bread. The Machatzit Hashekel 583:5 also says it is forbidden to throw food into the water.
- The Mateh Efraim 598:4 and Kitzur S"A 129:21. Eliya Rabba 596:3 writes based on kabbala that shaking out your pockets is like shaking off the kelipot (layers) that stick to us when we sin. Rabbi Eli Mansour points out that this is in no way a substitute for repenting, just a symbolic act to show that our repentance is sincere.
- Aruch Hashulchan 583:4 and Elef Hamagen 583:7. Rabbi Eli Mansour, Ketzeh Hamateh 598:7 and Yalkut Yosef Moadim 35 say that women aren't obligated to do tashlich and therefore shouldn't in order to prevent inappropriate mingling between genders.
- Yalkut Yosef Moadim page 35
- Ben Ish Chai Parashat HaNetzavim: 12 says to say it even on shabbat. Kaf Hachayim 583:31 says that is the custom in Yerushalayim. Shvut Yaakov 3:42 quotes the Shnei Luchot Habrit that it should be postponed, but disagrees and says to say it on shabbat, and quotes the Maharil (Teshuva 136) as saying the same. This is also the ruling of the Mateh Ephraim 598:4, Shaare Teshuva 583:6. The Elef Hamagen 598:11 says to postpone it. Rabbi Eli Mansour says this is the custom of the Brooklyn Syrian Community based on the psak of Chacham Baruch ben Chaim. See Yabia Omer 4:47 and Yalkut Yosef Moadim page 34 for a summary of the opinions on this matter.
- S"A 597:1, Chazon Ovadyah (Yamim Noraim, pg 180)
- Mishna Brurah 597:2 writes that even though usually on Shabbat and Yom Tov one shouldn't fast until after chatzot on rosh hashana it's permissible to allow the davening to extend past chatzot. Rav Hershel Schachter on yutorah.org (min 40-42) explains that this leniency is based on the fact that there are opinions in the Geonim who even permit fasting completely on Rosh Hashana.
- Chaye Adam 24:10, Sh"t Igrot Moshe 1:170, Rabbi Hershel Schachter in a shuir (min 43-45:30)
- Rav Moshe Shternbuch in Teshuvot Vihanhagot 2:268 quotes the the Vilna Gaon in his Sefer Maaseh Rav 207 that it is forbidden to cry, but also quotes the Arizal (brought down in Baer Heitev 584) that any person with a good soul would cry on Rosh Hashana. He claims that they aren't disagreeing, but rather that the Vilna Gaon is talking about crying from fear of the day of judgement and viewing the day as one of sadness and gloom. But if the tears come spontaneously from a longing for Hashem during the prayers, that it praiseworthy.
- Mishna Brurah 188:19 quotes the Magan Avraham 188:7 who says one doesn't repeat Birkat HaMazon and the Eliyah Rabba who argues and the Mishna Brurah leaves it unresolved. Shemirat Shabbat Kehilchata 57:7(4) quotes both opinions and in note 25 he concludes that one doesn't have to repeat benching because it's a Safek Brachot LeHakel. Similarly, Rabbi Hershel Schachter in a shuir (min 41-43) mentions that some achronim hold that one doesn't have to repeat benching because there is an opinion that one is allowed to fast on Rosh Hashana. When asked whether this would be considered Safek Deoritta Lechumra he explained that even if one forgot Yaaleh VeYavo one fulfills the Deoritta obligation of benching it just that one didn't fulfill the din derabbanan to have a meal for Yom Tov.