The best hospital, with modern equipment, excellent doctors (mainly from Ukraine and India) and an emergency section is the Tasakhtaa Global Hospital in Vuga, Stone Town: http://www.tasakhtaahospital.co.tz
Reception for making an appointment: +255 (0) 24 223 2341
Emergencies: +255 (0) 24 223 2222, if you need an ambulance +255 (0) 779 770 577
In the bigger villages at the coast, we have so-called Clinics run by qualified nurses, they can also administer antibiotics if needed.
Should you get sick, best contact the manager or reception of your place of accommodation, they should have the numbers of the nearest medical practitioner.
Pharmacies: Shamshu (best selection) and Izmir Pharmacies, both in Darajani/Stone Town, behind the fish/meat market building, and at Global Hospital of course
In the rural areas, there are small Dawa (Kishwahili for medicine) shops, but they are badly stocked.
I suggest that you bring an emergency travel kit (especially antibacterial ointment) which you could then donate to the local clinic, it will be much appreciated.
Zanzibar is not an official Malaria area anymore. It is up to you to take Malaria prophylaxis or not. Do bring Mosquito repellent and drink lots of Tonic water (Mosquitos hate the smell of chinine in the tonic water which your skin somehow gives off). If you drink alcohol, enjoy a gin tonic at sunset (gin also contains chinine = double dose of chinine) – the gin tonic is only to keep the Mosquitos away of course 🙂
In the hotel rooms, you have mosquito nets and staff usually spray the rooms at night when they get your bed ready.
YELLOW FEVER VACCINATION
According to the WHO, a yellow fever vaccination is NOT required for Zanzibar, except you arrive from a country with recent yellow fever cases. I have a letter that confirms this, pls email me on email@example.com and I gladly forward the letter to you.