Before your trip please check your vaccination document and make sure that all your routine vaccines are up-to-date. These vaccines include measles-mumps-rubella (MMR) vaccine, diphtheria-tetanus-pertussis vaccine and polio vaccine.
There are certain vaccinations (hepatitis A + B, thypoid and rabies) recommended, for further details please contact your doctor. Travellers that wish to go to Zanzibar will be required to have Yellow Fever vaccination. Tanzania in general does not require yellow fever vaccination, but if you have a connecting flight within Africa, a yellow fever certificate is required to enter Tanzania.
Additionally you want to consider malaria prophylaxis.
Please discuss the possibilities with your doctor.