Zanzibar (also known in Arabic as Zanjibar) is a beautiful Island to explore by the Indian Ocean. We first went to Paje beach, which is located in the southeast of Zanzibar, we stayed in a hotel that looked beautiful but was extremely dirty (Paradise Beach Bungalows). Make sure there is an air condition working in your room, it is very hot. Paje is a lagoon and it is famous for its kitesurfing and kiteboarding.
Visit The Rock Cafe which is on the Michamvi Pingue just 10 minutes away from Paje by car, the Cafe was voted to be one of the most beautiful restaurants around the world. Make sure you do early reservations as it is always fully booked. You can also try Upendo which is opposite of The Rock Cafe and offers great food and ambiance.
We then moved to Nungwi, which was recommended by most of websites and people who have been to Zanzibar. It is also known as Ras Nungwi (Ras in Arabic stands for head/tip) as it is located in the far north tip end of the island. We loved it, and stayed there for about a week, our hotel was amazing and is highly recommended (Mnarani Beach Cottages).
I would recommend you settle in one area in Zanzibar and enjoy the breathtaking scenery that you will never get enough of. The distance from one area to another is long, about an hour or an hour and a half from Paje to Nungwi.
Learn how to play Bao, check out the shops by the beach, have seafood, try Langi Langi (an Italian cuisine, with great food), meet their people, stay away from the beach boys who will always try to sell you something (and sometimes you can actually get yourself a good offer), watch the sunset, lay down on the sand and get a beautiful suntan.