Pythagoras also attended Thales's pupil's, Anaximander, classes in Miletus, whose ideas influenced Pythagoras' views. Around the year 535 BC Pythagoras went to Egypt where he visited many temples and led many discussions with the priests and even was accepted into priesthood.
Pythagoras settled in Crotone, a Greek colony in southern Italy, about 530 BC, where he founded a philosophical and religious school. Members of his society were divided into mathematikoi and akousmatics. The mathematikoi were the inner circle, they lived with the society all the time and had no personal possessions and were vegetarians. Pythagoras taught them and they had to obey strict rules. The akousmetics, the outer circle, came to the society only during the day and lived in their houses - they didn't have to be vegetarians and were allowed to have personal possessions.
Both men and women could become members of the society and soon Pythagoras had many followers. They were very religious and aimed for reforming political, moral and social life. They believed that the reality is mathematical in nature and that philosophy is the way to the spiritual purification - that is the time, when soul unites with the divine.
The school practised communalism and secrecy so it's difficult to tell which achievments are Pythagoras' and which are his students'. However, it is certain that the mathematikoi with their leader devoted themselves to mathematics and they introduced irrational numbers. However, it is unlikely that it's Pythagoras's achievement, because it was against his philosophy (he thought that all things are numbers and by numbers he meant a ratio of two whole numbers).
Pythagoreans tried to stay out of politics, but some sources say, that this order took part in a discussion against the war that Crotone declared on its neighbour, Sybaris. Two years later one of Crotone's nobles attacked the society.
After 500 BC the society has grown in number rapidly and took part in politics and also split into many fractions.
Pythagoras, often called the first true mathematician, is responsible for a huge development in mathematics, but didn't leave any written evidence of his achievements. His society worked in secrecy, which makes makes a mysterious figure out of him.
Although Pythagoras studied even and odd numbers and perfect numbers and triangular numbers he thought that every number has its own personality and ten was the best number of all - it was the sum of the first four integers (1+2+3+4 = 10) - and these written in a special way formed a perfect triangle.
Pythagoras is most famous for his geometry theorem - which was in fact known to the Babylonians 1000 years earlier, but he was probably the first one to prove it. He also knew that Earth was a sphere, but he thought it was in the centre of the universe.
However, Pythagoras was primarily a philosopher. He thought that one's soul is a number travelling through the world and experiencing reincarnation into different species until it eventually purifies and that all things were first made of form and not from material substance.
We don't know much about his death - he certainly escaped to Metapondium after the attack on his society and probably died here - according to some authors he committed suicide because of the attack.