Tanzania has two main dry spells, roughly June to October and January to February, and two rainy periods. The dry months are good for game viewing because animals gather at water; the green season is quieter and cheaper in many places.
The Great Migration moves around the Serengeti through the year. River crossings in the north are usually talked about for the middle of the year, but timing varies, so ask for current guidance when you book.
Kilimanjaro is climbed in most months. The drier months are popular; the wetter ones are quieter and colder, with more cloud. Zanzibar is warm all year with a few wetter weeks.