English is widely spoken in Tanzanian tourism and you'll manage perfectly without a word of Swahili. Learn twenty anyway. The reaction you get is out of all proportion to the effort.
Greetings come first, always
This is the cultural point hiding inside the language one. In Tanzania you greet before you ask. Walking up to someone and starting with a question is abrupt in a way it isn't in Europe, and the greeting isn't a formality, it's the thing itself.
Jambo is the tourist hello and everyone will use it with you. Habari? means 'what news?' and is the everyday one; the answer is nzuri, good. Mambo? is informal, and the reply is poa, cool. To an older person you say shikamoo, which is respectful, and they'll answer marahaba. That last one will get you a smile every single time.
The essential handful
Asante, thank you. Asante sana, thank you very much. Karibu, welcome, and also 'you're welcome'; you'll hear it constantly. Tafadhali, please. Ndiyo and hapana, yes and no. Samahani, excuse me or sorry. Pole, which means sorry in the sense of sympathy, said to someone who's stumbled or had a hard day. Hakuna matata is real, and really used.
Words you'll hear on safari
Pole pole, slowly, which is the entire philosophy of climbing Kilimanjaro. Twende, let's go. Simba, lion. Tembo, elephant. Chui, leopard. Duma, cheetah. Faru, rhino. Nyati, buffalo. Ndege, bird, and also aeroplane. Maji, water. Safari itself just means journey.
A couple that are useful in town
Bei gani?, how much? Ghali sana, very expensive, said with a smile, is the opening move in any market. Sielewi, I don't understand. Choo kiko wapi?, where's the toilet, which is the phrase everyone wishes they'd learned.
Pronunciation, briefly
Easier than it looks. Vowels are consistent: a as in father, e as in bed, i as in machine, o as in or, u as in rude. Stress nearly always falls on the second-to-last syllable. Say it as it's written and you'll be understood.
Why it's worth it
Because the moment you greet a porter or a waiter or a ranger in their own language, the conversation changes. You stop being a guest being serviced and start being a person who turned up. It costs an afternoon's practice.
Numbers, for the market
Moja, mbili, tatu, nne, tano is one to five; sita, saba, nane, tisa, kumi takes you to ten. Mia is a hundred and elfu is a thousand, which is the one that matters, because Tanzanian shillings come in thousands and elfu tano is five thousand. Knowing the numbers changes a market conversation more than any other twenty words here.
One warning about jambo
Jambo is the greeting Tanzanians use with tourists, and almost never with each other. It isn't wrong and nobody minds. But if you lead with habari or mambo instead, you'll notice the difference in the reply, because you've stepped over the line between being a visitor and making an effort.