To get an overview of the country and its history, culture etc, The Portuguese by Barry Hatton (an English journalist who has lived in Portugal for 20+ years) is a good place to start.
Things that are unique to Portugal and/or important in Portuguese identity include (but are obviously not limited to): fado, azulejos, Luís de Camões, saudade (a kind of longing or nostalgia), the Age of Discovery, and Portugal's historic rivalry with Castile/Spain. A book like the one listed above will touch on all of them.
Some other things that are important and/or unique in Portugal include the Portuguese language (the 5th/6th most widely spoken language in the world depending on how you count, and not as close to Spanish as most people assume), the Carnation Revolution (the 1974 army coup that overthrew the dictatorship and installed democracy, which is much celebrated) and Manueline architecture (a unique form of architecture tied specifically to the Age of Discovery that you can see in its most glorious form in Belém, just outside Lisbon).
Traditional and much-loved local cuisine includes bacalhau (cod fish), sardines (which are celebrated with preservative tourist shops in Lisbon), açordo (a fish/bread stew) and pastéis de nata (Portuguese custard tarts). Local alcoholic drinks include port wine (from Porto), vinho verde (green wine) and ginja or ginjinha (a cherry liquor, famous in Óbidos but widely available in Lisbon).
For history books on specific events, the best two, in my opinion, are This Gulf of Fire by Mark Molesky about the 1755 Lisbon earthquake, and Conquerers, by Roger Crowley, about the most dramatic 30-year period of the Age of Discovery.
Famous Portuguese (or proto-Portuguese) people and/or heroes that you could look up include (but are not limited to): Viriatus, Affonso Henriques, Nuno Álvares Pereira, Henry the Navigator, Bartholomew Dias, King Manuel the Fortunate, Vasco da Gama, Ferdinand Magellan (despite sailing under the Spanish flag when he proved that the world could be circumnavigated), Luís de Camões, the Marquês de Pombal, Fernando Pessoa, António Salazar, Amália Rodrigues, José Saramago and Cristiano Ronaldo.
Hopefully that will get you started!
(P.S. To our Portuguese friends on the forum, I hope this is a fair summary!)