The Path Between the Seas by David McCullough (1977) is the definitive account of the Panama Canal's construction — the failed French attempt under Ferdinand de Lesseps, the yellow fever and malaria that killed tens of thousands of workers, the American takeover under Theodore Roosevelt, and the engineers Stevens and Goethals who finally made it work. It is one of the great works of American narrative history and will change how you look at the locks.
The Tailor of Panama by John le Carré (1996) is a novel set in Panama City in the years around the handover of the Canal Zone, following a disgraced British spy and his dealings with a local tailor who manufactures intelligence. Le Carré acknowledged the debt to Graham Greene's Our Man in Havana, and the Panama City atmosphere — heat, money, political intrigue, expat community — is rendered precisely. Read it on the plane or in the Casco.
A Guide to the Birds of Panama by Robert S. Ridgely and John A. Gwynne is the standard field guide for birding in Panama and is essential preparation for a visit to Pipeline Road or Soberanía. The second edition covers all species recorded in Panama and is the reference used by professional guides.
Birds of Central America by Andrew Vallely and Dale Dyer (2018) is a more recent and comprehensive regional field guide covering all of Central America, with better illustrations for some species than the Panama-specific guide. Useful if you are travelling beyond Panama or want a single reference for the region.