The most obvious example of Puritan thinking is that society has a right to control what other people do. Obviously, they frowed on adultery and sin in general, but they also felt the society's role was to punish offenders.
1. The theocracy of Puritanism involves the control over every aspect of people's lives, religious and civil. e.g. The governor interviews Hester regarding her child's religious upbringing.
2. The concept that there is no retribution for sin. No matter what Hester does, she is never forgiven her sin of adultery. This is why she picks up the letter and replaces it when she returns to the Massachusetts community.
3. The falsity of such a somber religion. e.g. People hide their secret sins for fear of being punished and ostracized.
Three traits of Puritanism reflected in Hawthorne's novel might include the following:
1. The assumption that politics, government, and law should be heavily influenced by the Protestant religion.
2. The assumption that the private lives of individuals, including and perhaps especially their sexual lives, should be heavily influenced by the Protestant religion.
3. The assumption that one's fate in the hereafter was far more important than any temporal, temporary pleasures in the here and now.
Supplying you with specific quotations and chapter references wouldn't be engaging in a discussion, but maybe this will be helpful. The entire novel is reflective of Puritan thinking as it affected the lives of Hester, Arthur, and Pearl. Each chapter in the novel is rich with references to and examples of Puritan thinking in the setting of the novel. For instance, the continual references to sin, shame, punishment, and spiritual redemption all reflect Puritan religious beliefs, as do the references to modest behavior. If you will recall, Hester's dressing Pearl in bright colors was an act of rebellion, as the Puritan code called for dressing in plain, dark colors; the word "gray" is mentioned in the novel numerous times. I would suggest you skim read various chapters looking for these key words. They will lead you into passages that reflect the thinking of the Puritans. Good luck!
Keep in mind that Nathaniel Hawthorne was writing during the Era of Romanticism, and not during the Puritan Era. Therefore, the novel is showing a picture of Puritanism filtered through the lens of an author who did not believe in the Puritan way of life. For example, Puritans had a very defined view of what was an acceptable faith and view of God. The Romantics believed a person’s view of God came more through personal intuition, and should not be imposed on another.