Short answers, pulled from the story.
Vasco da Gama arrived in Calicut on the western coast of India in May 1498. He departed Portugal on the 8th of July 1497 with four ships and 170 men, rounded the Cape of Good Hope, traveled up the East African coast, and crossed the Indian Ocean with the help of a local pilot.
Henry the Navigator was a Portuguese prince appointed governor of the Order of Christ in 1420. He funded and directed sustained coastal exploration along West Africa, sponsoring voyages down the Mauritanian coast and gathering merchants and shipowners to invest in new trade routes. He directed Portuguese exploration until his death in 1460.
The Treaty of Tordesillas, signed in 1494, drew a north-south line 370 leagues west of the Cape Verde islands. Portugal claimed all non-Christian lands east of that line; Spain claimed all lands to the west. The precise boundary remained disputed because accurate longitude measurement was not yet possible, and the disagreement continued until 1777.
Pedro Álvares Cabral made landfall on the Brazilian coast in 1500 while sailing west across the Atlantic to follow the same wind-favorable route to India that da Gama had used. The discovery was probably accidental. Cabral found the land rich in pau-brasil, the reddish hardwood that later gave the country its name, and recommended settlement to the Portuguese king.
The Battle of Diu in 1509 was a sea battle in which Portugal defeated a coalition including the Ottoman Sultan Beyazid II, the Sultan of Gujarat, the Mamluk Sultan of Cairo, and the Venetian and Ragusan Republics. The victory caused the Ottomans and Egyptians to withdraw their navies from the Indian Ocean, leaving Portugal in commercial control of those waters for nearly a century.
Francisco Zeimoto, António Mota, and other Portuguese traders arrived in Japan in 1542, landing at Tanegashima. According to Fernão Mendes Pinto, who claimed to be present, the Japanese were immediately impressed by European firearms and began making them on a large scale.