Staying in Stone Town for a few nights is a must. Getting lost around the streets is great fun (but see how long you can handle the often aggressive store owners), and there are plenty of bargains to be found. The nighttime food markets are also a treat - just try not to think about hygiene too much.
The northern beaches are beautiful - turquoise water, pale white sand, and a surprisingly low number of tourists!
Quick Tips:
Spice tour is one of those things everyone does, so you'll do it anyway, but its probably not worth it. Depends what you want, I guess.
Best Way To Get Around:
Once at each location, go on foot. Otherwise, either locally arranged tour or local transport.