These two days constitute a major holiday, when most forms of work are prohibited. On the preceding nights, women and girls light candles, reciting the appropriate blessings, and we enjoy nightly and daily festive meals, accompanied by the Kiddush. We don't go to work, drive, write, or switch electric devices on or off. We are permitted to cook and to carry outdoors (unless it is also Shabbat).
The first day, Shemini Atzeret, features the prayers for rain, officially commemorating the start of the Mediterranean (i.e., Israeli) rain season, and the Yizkor (prayer supplicating G‑d to remember the souls of the departed).
We no longer take the Four Kinds, and we no longer mention Sukkot in the day's prayers; in the Diaspora, however, we do still eat in the sukkah (but without reciting the blessing on the sukkah).
The highlight of the second day....well those can be found by clicking here.