Continue learning the Gemara until the Mishna דף מח
Quickly review the גמרא דף מז: “וסבר ר’ יהודה until “מאי הוי עלה (four lines from bottom of the Amud)
Learn תוספת מז. “לינה
Questions:
How many nights is one obligated to stay in Yerushalayim during Sukkos?
List the different requirements for לינה
Continue learning the Gemara until the Mishna דף מח
Questions:
Why did ר’ יהודה only list four out of six items in which Shemini Atzeres is different from the rest of Sukkos?
Why doesn’t the braissa list the difference of לינה?
Learn “רש”י ראש השנה דף ד: “פז”ר קש”ב
Note Rashi gives a different explanation in what רגל בפני עצמו means than he does in our Gemara.
Learn “תוספת סוכה מח. “רגל who asks a question on Rashi and brings different opinions for the explanation of רגל בפני עצמו
Learn ריטב”א who discusses many of the above questions.
In Conclusion:
Is Shemini Atzeres an entirely new Yom Tov or is it an extension of Sukkos?
If in shemoneh esrei or birkas hamazon a person says “וביום חג הסוכות instead of וביום שמיני עצרת does he have to repeat?
What is the correct wording used in shemoneh esrei when mentioning Shemini Atzeres?
See רש”י דף מז: “ברכת המזון
Learn שולחן ערוך אורח חיים ס’ תרס”ח, א (until “ובחוץ לארץ”)
https://hebrewbooks.org/pdfpager.aspx?req=14327&st=&pgnum=408
See מגן אברהם ס”ק א and learn segments of the ט”ז ס”ק א
Learn משנה ברורה ס”ק ב & ג
https://hebrewbooks.org/pdfpager.aspx?req=60391&st=&pgnum=274