I think you can find a lot of information on the Wikipedia link you have provided, especially when you start checking out the resources on separate events during that time.
But, when looking for the conditions Peter Stuyvesant encountered before he came to North America you should look into the colony known as "New Netherland". A hostile place with conflicts with Native Americans, New Sweden to the south and New England in the East. The Dutch West India Company was supposed to provide protection, but were more interested in the sugar plantations in Dutch Brazil, the Dutch Antilles and the slave Trade. It wasn't until 1654, after were forced out Dutch Brazil by the Spanish, that they belatedly started to focus again on North America.
And you should look into the mess he inherited from his predecessor Willem Kieft who was called back (fired) by the board of Directors of the Dutch West India Company, after he waged a bloody war between the Native Americans and the Dutch colonists known as Kieft's War.
New Amsterdam was a growing and prosperous Colony up until these wars after which many colonists fled and the trust in the Dutch West India Company to create a safe city was lost. Peter Stuyvesant's period was a relatively calm period where the population of the city of New Amsterdam had grown from 2.000 to 10.000
Peter Stuyvesant's legacy was made big because he made the city of New Amsterdam safer by building a big wall in the place now known as "Wallstreet". He laid a base street plan of Manhattan in order to best defend the place. He was forced by the Dutch West India Company to accept people from different religions and backgrounds in his city laying the cultural foundation to New York City. And mostly he is know for surrendering New Amsterdam peacefully over to the English after which the name was changed to New york.
In October 1664 every Dutch citizen in New York, including Stuyvesant,
had to swear allegiance to the British king. During Stuyvesant's
government the population of New Netherland had increased from 2,000
in 1647 to 10,000. The West India Company summoned Stuyvesant to
Amsterdam and charged him with neglect, but he criticized the
shortcomings of the Company policies and support. After the second
Anglo-Dutch War of 1665-67, Stuyvesant returned to New York and lived
on his farm until his death in 1672.