Zanzibar Island, a.k.a., The Spice Island, was an important stop in the Spice Trade centuries ago. Today, it is one of the few places in the world where saffron is produced, and many other Middle Eastern/Asian spices cardamon, ginger, cloves, nutmeg, etc. are grown here. For $10-$15 you can go on a spice tour, which winds you around the island, showing you how anise licorice grows; letting you sample some of the exotic fruit grown on the island; and allowing you to tour the beautiful plantations.
Jozani Forest
has excellent nature trails, featuring some very exotic and large trees. Even more interesting, though, are the Red Colobus Monkeys that live here. Native to the Island, these monkeys are now nearly extinct. They are very curious and playful and will likely pose for a picture. The entry fee 8$ also include an optional visit to a beautiful mangrove forest which is highly recommended.There are a number of historically important and frankly, just plain beautiful buildings in Stone Town, like The House of Wonders and The Arab Fort. It is easy to arrange a simple walking tour with a local guide who can teach you some history.
The market in Stone Town is one of the largest, most vibrant open-air markets anywhere. Here, you can find several varieties of bananas, "elephant garlic" unique to the island, the largest avocados you'll probably ever see, and more. Prices are extremely reasonable. Even if you have no intentions of purchasing food, the spectacle alone is worth a visit.