Short answers, pulled from the story.
Plato was born between 428 and 423 BC into an aristocratic Athenian family. His mother Perictione descended from Solon, the statesman credited with laying the foundations of Athenian democracy.
Ancient writers record that Plato had two brothers named Glaucon and Adeimantus who both appear in his Republic dialogue. He also had a sister named Potone and a half brother named Antiphon.
After the death of Socrates Plato remained in Athens for roughly three years before leaving around 396 BC to study in Megara with Euclid of Megara. He returned to Athens around 394 BC or earlier where he spent time studying mathematics with Archytas of Tarentum and others.
Plato founded his philosophical school the Academy near the sacred olive grove of Hecademus around 383 BC. At first the property consisted only of a house with a garden while later it took part in an open area for study of philosophy and mathematics.
Plato visited Syracuse three times starting around 385 BC when Dionysius I ruled. He returned to Syracuse again likely early in 366 BC at request of Dion and then a third time in 361 BC staying over winter until 360 BC.
Plato's theory of Forms aims to solve what is now known as the problem of universals by denying reality of the material world considering it only an image or copy of real world. There are two kinds of things: apparent world of material objects grasped by senses which constantly changes, and unchanging unseen world of Forms grasped by reason.