Africa: A Biography of the Continent by John Reader — The best single-volume history of sub-Saharan Africa, placing Tanzania's ancient human history (including Olduvai Gorge) in continental context. Essential reading before any serious trip to the region.
Through a Window: My Thirty Years with the Chimpanzees of Gombe by Jane Goodall — Goodall's landmark account of her research at Gombe Stream, one of the finest records of sustained wildlife observation ever written. Particularly relevant before visiting Gombe.
The Shadow of Kilimanjaro by Rick Ridgeway — A travel memoir covering the landscape between Kilimanjaro and the coast on foot. Vivid and unpretentious.
Scramble for Africa by Thomas Pakenham — The history of European colonisation, including German East Africa (today's Tanzania), told in energetic narrative form. Explains a great deal about what you will see on the ground.
Serengeti Shall Not Die by Bernhard and Michael Grzimek — The book that helped establish Serengeti as a protected park, written in the 1950s. Still worth reading for its portrait of the ecosystem before mass tourism.
The Zanzibar Chest by Aidan Hartley — A foreign correspondent's memoir partly set in East Africa during the turbulent 1990s. Sharp and honest about the region.
For practical guidance, the Lonely Planet Tanzania guide is reliable and regularly updated. The Bradt guide offers more depth on off-the-beaten-path destinations.