Shulchan Aruch Chapter 598:
No Title (1 Paragraph)
Note The REMA is in brackets []

 1.  If Rosh Hashanah falls out on Shabbat we say ...צדקתך in Mincha (מנחה, Afternoon Service). [There are those who say that you do not say it.]

 
Chapter 599:  No Title (1 Paragraph)

 1.  The eve of Rosh Hashanah (ליל ר׳ה) that falls out on [מוצאי שבת [1 we say ...ותודיענו.

 
Chapter 600:  An egg that was made on Rosh Hashanah and the laws of Kiddush
(3 Paragraphs)

 1.  If an egg was made on the first day of the festival and similarly whatever was caught with it and whatever is touching it is forbidden on the second day [2].  [This was already explained in 513:5 and chapter 515.] 

 2.  In Kiddush of the second night [you should wear a new garment or] place a new fruit before you and say שהחיינו.  And if you can find a new fruit in spite of all this you should say שהחיינו.

 3.  If the fist day falls out on Shabbat, we say the שהחיינו over the Shofer on the second day.  [There are those who say that you should (also) say it (on the second day) even if the first day is on a weekday and such is the custom in these areas.]

 Chapter 601:  The arrangement of the second day of Rosh Hashanah (2 Paragraphs)

 1.  On the second day [we pray like on the first day] and we read from “...והאלקים” (Genesis 22:1-4) until the end of the portion.  For Maftir we read like the day before (Numbers 29:1-6). We conclude with the following found in Jeremiah “כה אמר ה׳ מצא חן במדבר” until “הבן יקיר לי אפרים”(Jeremiah 31:1-20).  And the night following Rosh Hashanah (מוצאי ר׳ה) we make Havdalah like after Shabbat but we don’t make the blessing over the spices and on the fire.

 2.  Even in the land of Israel we make Rosh Hashanah two days.

 Chapter 602:  The arrangement of the days of Repentance (1 Paragraph)

 1.  All the days between Rosh Hashanah and The Day of Atonement (Yom Kippur), we indulge in prayers and supplications.  [And we say אלוקים מלך morning and evening except on Shabbat.  Even if a Brit Milah (circumcision) occurs where you don’t say [תחנון [3, nonetheless we say אלוקים מלך.  And we confess (say ודוי) three times every day before day break (עלות השחר) except on the day preceding the Day of Atonement where we only say it one time.  We don’t put someone in חרם (the ban) and similarly we don’t allow someone to swear before the court until after the Day of Atonement.  We don’t sanctify the new moon until after the Day of Atonement.  On the Sabbath between Rosh Hashanah and The Day of Atonement, the custom is to say Tzidkascha. ]
 
Chapter 603:  No Title (1 Paragraph)

 1.  Even one who is normally not concerned about using bread baked by a gentile [4], it is necessary to be careful on the ten days of repentance.  [Every one should seek out and inspect his deeds and to turn them back (from sin) in the Ten Days of Repentance.  When there is a doubt if a sin was committed (ספק עבירה), more repentance is necessary than a unquestionable sin since something is more engraved (in memory) when one knows about it then when one is unaware.  For this reason, the [אשם תלוי קרבן [5 must be more valuable than a חטאת (sin offering). ]


END OF HALACHOT ROSH HASHANAH
Translated by Jay Dinovitser 7/06 shulchanarach.com
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[1] The night just after the Sabbath ends.
[2] M.B: On other festivals it would be permitted to use on the second day but not on Rosh Hashanah.  The reason is that on R.H. both days are sanctified as one.  On the rest of the festivals, one of the days is a regular day and one is a festival day on a strictly Biblical basis.    
[3] A supplication for forgiveness said during the weekday where you fall on your face.  See your prayer book for more information.
[4] Some Jews have the custom to not eat bread baked by a gentile and some do not.  Food baked by non Jews is a completely different issue and is strictly forbidden for most foods unless a Jew turned on the fire and supervised as verified by the Kosher certificates in front of the Kosher eateries and Kosher symbols on food.  
[5] Guilt offering given in the above situation at the time of the temple.