Most American discussions of church and state begin from a political point of view: conservative or liberal, left or right, republican , democrat. In today's society, these terms are ambiguous at best, and often misleading or used in a derogatory sense, so I am going to avoid them.
Instead, in approaching the question “What is the role of government?” I will answer from an unashamed, hopefully unambiguous, Christian perspective. Admittedly there are many of these, also, so I will further limit this to a discussion of Christian principles based on the Bible, rather than specific doctrines or agendas.
God's View of Humanity
By God's definition, we are all rebellious sinners. Created to worship Him, we have instead refused to submit and chosen to worship ourselves and the things we have created. God's holiness demands an eternity of punishment for this sin. Nonetheless, the cry of God's heart is that all men be saved and come to the knowledge of the truth. In His mercy, He sent Jesus to pay the price for our sin, and the Holy Spirit to convict us of our sin and guide us into all truth.
God does not, however, force anyone to worship Him. From beginning to end, the Bible can be viewed as an extended love letter from God, pleading with man to repent and return to their Creator. It can also be viewed as the words of a wise Father, warning his children not to play with various forms of fire because they cause burns. While He may orchestrate circumstances to reveal our sin, in the final analysis, the choice is always left to the individual.
The Role of the Church
Among those who have repented and been saved, our role is two-fold. First, we are to love God with all our heart and soul and strength and mind. Second, we are to love our neighbors as ourselves. Implicit in this summary of the law, and more explicit in other passages, is that God's desires should be our desires, and His methods, our methods. We are specifically instructed to speak the truth with love, help those in need, and preach the gospel to all creation.
The Church in Practice
The church characterized in the New Testament largely, but imperfectly, achieved these goals. The gospel was preached. God was worshipped above all else, and many were drawn into the kingdom. Those in need were helped in love, the rebellious heard hard truths in love, and all were allowed to freely choose.
Since that time, however, the historical church has generally failed in its assigned task. Instead of pointing the way to Christ, church leadership has interposed themselves between Jesus and man. Instead of allowing the Holy Spirit to do his job of convicting and guiding, man has arrogantly assumed this position.
Around 312 AD, Emperor Constantine converted to Christianity, and subsequently put the weight of the government behind the church. Thus began the practice of state enforcement of the doctrines determined by the established church, leading to centuries of wars.
The American Church, 2008
Things have not changed much in the United States. With the Salem witch trials a notable exception, we no longer execute people for their religious beliefs. However, the freedom of religion defined in the First Amendment is, in practice, a myth.
American Christians typically approach government from one of two perspectives. Desiring to help those in need, they use the government to redistribute wealth. Others, motivated to see repentance from sin, use the government to define and enforce moral behavior. Both positions use the coercion of the state to enforce religious practice. Neither position draws people to Christ, and in fact, interferes with the work of the Holy Spirit. In addition, it allows Christians to avoid their personal obligation to speak the truth with love, help those in need, and preach the gospel to all creation.
God gives each individual the free will to choose to repent and follow Him, or to continue on the path to hell. Government should do the same. The language of the First Amendment in this regard is God-given:
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof ;
The failures of the church to convert society to Christianity should not become the job of the state. The role of government should be limited in this regard to that defined by Thomas Jefferson:
Of liberty I would say that, in the whole plenitude of its extent, it is unobstructed action according to our will. But rightful liberty is unobstructed action according to our will within limits drawn around us by the equal rights of others. I do not add 'within the limits of the law,' because law is often but the tyrant's will, and always so when it violates the right of an individual. letter to Isaac Hall Tiffany, Esq., April 4, 1819
If Christians truly desire to draw others to Christ, they will not force society at large to fulfill their responsibilities. Moral laws do not change people, they only create self-justified hypocrites or criminals. Neither do laws save people. God changes people, and God alone, through Jesus Christ and the Holy Spirit, can save people.