Good questions and because the names of the holidays are so different and often unfamiliar here are the answers:
Sukkot - Is the week long festival where Jews "dwell" in "booths". There are all kinds of them but the main aspect of each is that one must be able to see the stars through the roof. The "dwelling" for some means they eat, sleep and study in their Sukkah (booth). For many of us we just eat there. The rational behind the Sukkah is to remind us of the wanderings in Sinai as well as reminding us that life...is fragile. Besides it's a great fall event for family and friends and kids get a kick out of decorating the Sukkah.
Shemini Atzeres - This is a separate holiday "attached" to Sukkot. It gives us "one more day" to savor G-d's blessing. Rashi, the great Biblical commentator, explains that atzeret is an expression of affection, as would be used by a father to children who are departing from him. The father would say, "Your departure is difficult for me, tarry yet another day." The Jewish people prayed and brought offerings all the days of Sukkot so that the 70 nations of the world would have rain in the coming year. The Torah and the Almighty keeps us one more day for a special holiday to make requests just for ourselves. That's Shemini Atzeret.
If you want "Chapter and Verse" consider Exodus 33:12 - 34:26: Moses pleads to the Almighty to "make known to me Your ways." The Almighty commands Moshe to carve two stone tablets to replace the Tablets that Moses destroyed bearing the 10 Commandments. Moses carves them and ascends Mt. Sinai. The Almighty descends in a cloud and reveals to Moses the 13 Attributes of Divine Mercy which are constantly repeated in the Yom Kippur prayers. Moses asks the Almighty to "forgive our transgressions and make us Your Heritage". The Almighty responds that He shall seal a covenant with us. The Almighty then warns the Jewish people against idol worship (idolatry is believing that anything other than the Almighty has power). The reading ends with the Almighty commanding us to keep the Festivals -- Pesach, Shavuot and Sukkot.
There you have it my friends. May you have a wonderful holiday season.
And, Shabbat Shalom!