Matthew 22:15–22 is the key passage setting forth the basic text on separation of church and state. In this passage the Pharisees and Herodians faced Jesus with a loaded question (15–17). Involved was their messianic concept which forbade payment of taxes to a pagan power. To answer categorically either way would have involved Jesus in trouble with the Romans or the Jews. Jesus did neither (18). The coin testified to the Jews’ subservience to and dependence upon the State (19–20). They also recognized their relationship toward God. Jesus pointed out their obligations to both God and the State (21).
The principle of the separation of church and state does not mean that the two have no relationship whatever. Jesus recognized the existence, rights, and functions of the State (Matt 22:15–21). The early Christians in missionary work utilized roads and sea lanes provided by the State (cf. Paul’s travels, Acts 13–16; 27). On occasion Paul accepted or called for the protection of the State (Acts 18:12ff; 21:27ff; 22:25ff; 25:10–12). At the same time Christians were exhorted to submit to the authority of the State (Rom 13:1–7; 1 Peter 2:12–17). Even when persecuted by the State they were to endure it willingly as a testimony unto the Lord (1 Peter 3:14–15). In the peaceful existence of an orderly society they were to carry on their spiritual work (1 Tim 2:1ff).
Church and state are mutually related in the normal events of life. The state provides a proper atmosphere in which the churches carry on their work (cf. fire and police protection, national security, postal service, and general stability in society). In turn the churches endeavor to produce through the gospel the type of Christian character conducive to a well-ordered society.
But church and state are also mutually exclusive. Neither shall endeavor to control the other or to use it in the discharge of its separate responsibility. The church shall not seek to achieve its spiritual goals through political power (cf. Matt 4:8–10; John 6:15). Nor shall the state commandeer the church for political ends (Acts 4:19). No religion shall be favored above another. The state shall not levy taxes upon strictly religious property, nor shall any church receive tax funds to be used in the performance of its spiritual, educational, and healing ministry (cf. 1 Cor 16:1ff). The church shall be free to determine its own form of worship, faith, government, membership, and missionary outreach. But such shall be carried on within the framework of the laws of the state.
Working out Salvation with Fear and Trembling 
I have a post up about James Madison’s views on the separation of church and state that may interest you.