In Medieval Philosophy: An Introduction, Frederick Charles Copleston offers a precise account of Thomas Aquinas's five ways of proving the existence of God. Here's an excerpt from Chapter 6, "St. Thomas Aquinas":
"Aquinas, in the Summa Theologica, gives five ways of proving God's existence. First he argues from the fact of motion (which does not mean simply locomotion, but, as with Aristotle, the reduction of potentiality to act) to the existence of a first mover. This argument is based on Aristotle's argument in the Metaphysics. Secondly, he argues that there must be a first efficient cause; and, thirdly, that there must be a necessary being. We see that there are at any rate some beings which do not necessarily exist, for there are beings which begin to be and cease to be. But, these beings (contingent beings) would not exist, if they were the only type of being; for they are dependent for their existence. Ultimately there must exist a being which exists necessarily and is not dependent. The fourth argument proceeds from degrees of perfection observed in the world to the existence of a supreme or perfect being; and the fifth argument, based on the finality in the corporeal world, concludes with asserting the existence of God as cause of finality and order in the world. In these proofs the idea of dependence is fundamental, being successively applied to the observed facts of motion, efficient causality, coming into being and passing away, degrees of finite perfections, and lastly finality. None of the proofs were entirely new; nor did Aquinas think they were new. He was not writing for atheists but was engaged in showing the rational foundation of faith as a preliminary to treating of theological matters. The only proof which he develops at any length (in the Summa contra Gentiles) is the first, namely that from motion."