Everything below is the shape of the thing. The requirements change, they depend on where you've been as well as where you're going, and only a travel clinic with your records in front of them can tell you what applies to you. Go six to eight weeks before you fly, because some things need time.
Yellow fever, and the detail people miss
The requirement usually turns on your route rather than your nationality. Travellers arriving from, or transiting through, countries with yellow fever risk are generally asked for a certificate, and travellers coming straight from Europe or North America generally aren't. A connection through an African airport can change the answer, which catches people out. Check the current position with your clinic and the WHO travel advice, and carry the certificate if you have one.
Malaria
Present across most of the country, including the northern parks and Zanzibar, and the single most important health thing on your trip. Preventive tablets are standard advice and which one suits you is a medical decision, so that's a clinic conversation. Alongside the tablets: repellent with DEET in the evenings, long sleeves and trousers after dusk, and sleep under the net the camp provides. The mosquito that carries malaria bites at night, which is why evening discipline matters more than daytime.
And if you get a fever within a few weeks of coming home, tell the doctor you were in East Africa. That sentence is the one that gets you tested.
Routine vaccinations
A clinic will usually review the standard ones and look at hepatitis A, typhoid, tetanus and, depending on your trip, rabies and hepatitis B. Rabies comes up more than people expect because treatment after a bite is harder to get at distance from a city.
Food, water and the obvious
Drink bottled or filtered water, which every camp provides, and skip ice in places you don't trust. Most travellers are fine. See food and water for the rest.
What to bring in the bag
Your own regular medication in its original packaging with a copy of the prescription, rehydration sachets, something for an upset stomach, painkillers, antihistamine, plasters, antiseptic, high-factor sunscreen and a good insect repellent. Camps carry first-aid kits and your guide will have one in the vehicle, and neither is a substitute for your own tablets.
The altitude footnote
Ngorongoro's rim is above 2,200 metres and the crater lodges are cold and high enough that a few people feel a little breathless on the first night. It's nothing like Kilimanjaro, but if you're going on to climb, read the climb's health article too.
Travelling with children
Malaria prevention for children is a specialist conversation and the options differ by age and weight, so book the clinic appointment earlier than you would for yourself. Everything else is the usual: repellent, covered arms and legs at dusk, nets at night, and relentless hand washing. There's more on the rest of it in safari with children.