Short answers, pulled from the story.
Constantine's formal conversion is placed in 312, almost universally acknowledged by historians. He was baptized only on his deathbed in May 337, reportedly by the Arian bishop Eusebius of Nicomedia, having spent most of his adult life as a catechumen.
The Edict of Milan was issued in 313 by Constantine and Licinius. It granted religious tolerance to all faiths, returned property confiscated from the Church during the Great Persecution, and raised the standing of Christianity within the empire without making it the official state religion.
Eusebius of Caesarea records that Constantine saw a cross-shaped trophy of light above the sun at midday, accompanied by the words "By this conquer." Lactantius, writing earlier, describes only a dream the night before the battle instructing Constantine to mark a heavenly sign on his soldiers' shields. The exact symbol remains disputed among scholars.
Constantine summoned the First Council of Nicaea in 325, making it effectively the first ecumenical council in Christian history. It was the first major attempt to define Christian orthodoxy for the entire Church, and it dealt primarily with the Arian controversy over the nature of Christ's relationship to God the Father.
Constantine funded the Church directly, had basilicas built, granted clergy tax exemptions, promoted Christians to high offices, returned property seized during Diocletian's persecution, and commissioned fifty bound volumes of scripture for the churches of Constantinople in 331. He also built a new capital city, Constantinople, designed with Christian architecture from the outset.
The Constantinian shift is a term popularized by the Mennonite theologian John H. Yoder to describe the integration of imperial government with the Catholic Church beginning with the First Council of Nicaea. The historian Peter Leithart has argued there was a brief Constantinian moment in the fourth century but disputed that a permanent epochal shift occurred.