How did Puritans view sin, guilt, crime, and adultery?

One of the core ideas of Puritanism was the belief in Original Sin, that all humans are born condemned to spend eternity in hell. The exceptions were the few elect--those souls that God had chosen at birth to save from hell. This is predestination, the idea that people's fates are decided before birth; there is nothing people can do to change their fate. 

The Puritans constructed their society to be a "city upon a hill," or a model of Christian behavior, so they believed they had to punish infractions, whether religious or otherwise, quite harshly. The Puritans believed that any act against God was a heinous crime, and they punished crimes quite severely and publicly. Not attending church on Sunday, for example, could be punished by whipping. Other transgressors were placed in the stocks, where they were subject to public ridicule. The pillory, a post about 15 feet high, was used as punishment for crimes such as arson, beating one's wife, and forgery or cheating. The idea was to shame people into behaving by exposing their sins in public.

Sexual crimes were punished with particular severity; for example, sodomy was often punished by execution. In The Scarlet Letter, Hester Prynne must wear an "A" on her clothing as a badge of shame for having committed adultery. She is also largely shunned from the rest of society. In reality, some Puritans who committed adultery, seen as a grave sin and a violation of the Ten Commandments, were whipped. Some women who committed adultery were executed, as adultery was declared a capital crime in the Massachusetts Bay Colony. Men who committed adultery were often whipped but were not treated as harshly as women who committed this crime. This society, in which people stressed moral perfection, resulted in feelings of guilt as many people worried if they were worthy of God's grace. 

The Puritans believed that their church had entered into a covenant with God.  In this covenant, they pledged to act correctly in return for God's special favor (on Earth).

Because of this, they saw sin, crime, and adultery as violations of their covenant with God (and the covenants they made with each other to act in accordance with God's will).  These things had to be punished by the leaders of the church community so that God would not punish the society as a whole.

The Puritans did not, however, see sin as something that would cause people to be damned.  People were damned or saved based on predestination and nothing they could do would change their fate.

How did Puritans define guilt, sin, crime, and adultery in their religion and government?

The Puritans were Calvinists. They believed in predestination—that God pre-ordained some people to go to Heaven. God showed his favor on the Elect, the Puritans believed, by blessing them with good families and wealth here on Earth. In order to demonstrate one's membership in this group, it was important to be thrifty, hardworking, and virtuous. Puritans managed local affairs through the town meeting, where all male property owners could voice an opinion. Compared to the southern colonists, the Puritans were quite democratic. Puritans expected godly leaders and held them and each other to high moral standards. Neighbors were encouraged to tattle on each other for crimes such as blasphemy and breaking the Sabbath. Puritans valued the opinions of others; those who did not care were viewed with skepticism.

Crime, to the Puritans, was anything that broke the Bible's commandments. Laziness was also taboo in this world. Puritans used shame as an effective tool to punish wrongdoers. Puritans used stocks as public shaming, though serious offenders could expect to spend time in jail. The Puritans also hanged people for murder and witchcraft. Guilt came with one's awareness of sin, and all of mankind was guilty due to original sin. While adultery was not a large enough offense for a jail sentence, it did make one an outcast from the community, as well the local church, which served as the religious and social hub of the community.

The Puritans thought that sin was a result of the Original Sin, or the fall of Adam and Eve from God's grace in the Garden of Eden. They thought that, as a result of this sin, humans were sinful and guilty. While only God could choose the "elect," or the saved, in Puritans' eyes, they thought that people should live as if they were saved and stay away from crimes to show that they were among God's favored elect. Adultery was a particularly evil sin in their eyes, as men and women were supposed to pledge to be faithful to each other in marriage, replicating the covenant that they had to be faithful to God.

The Puritan government was organized as a theocracy. Church and government were intertwined, and, originally, only church members could serve in the government. Puritan religion and government were characterized by a belief in predestination—that God chose whom to save.

The first part of your question has been answered before on this site.  I'll confine my answer to the second part.   Please follow the second link to find a very good answer for the first part of the question.

Puritan government was theocratic.  That means that it was ruled on the basis of religious belief.  The Puritans believed that God would punish their society if it allowed wrong actions and/or beliefs to exist within it.

Therefore, only Puritan church members were allowed to vote.  The leaders of the church were also the leaders of the government (this is within towns, not within a whole colony).  They enforced religiously based laws and they used tax money to pay for Church buildings, minister salaries, etc.