You find Aquinas’s Five Ways in Summa Theologiæ 1a 2,3. I will here concentrate on the first three.
In the First Way, Aquinas goes from the fact that there is motion in the world to the existence of a “First Mover.” His argument is as follows:
It is certain, and evident to our senses, that in the world some things are in motion. Now whatever is in motion is put in motion by another, for nothing can be in motion except it is in potentiality to that towards which it is in motion; whereas a thing moves inasmuch as it is in act. For motion is nothing else than the reduction of something from potentiality to actuality. But nothing can be reduced from potentiality to actuality, except by something in a state of actuality. Thus that which is actually hot, as fire, makes wood, which is potentially hot, to be actually hot, and thereby moves and changes it. Now it is not possible that the same thing should be at once in actuality and potentiality in the same respect, but only in different respects. For what is actually hot cannot simultaneously be potentially hot; but it is simultaneously potentially cold. It is therefore impossible that in the same respect and in the same way a thing should be both mover and moved, i.e. that it should move itself. Therefore, whatever is in motion must be put in motion by another. If that by which it is put in motion be itself put in motion, then this also must needs be put in motion by another, and that by another again. But this cannot go on to infinity, because then there would be no first mover, and, consequently, no other mover; seeing that subsequent movers move only inasmuch as they are put in motion by the first mover; as the staff moves only because it is put in motion by the hand. Therefore it is necessary to arrive at a first mover, put in motion by no other; and this everyone understands to be God.
Here we need to clarify a few things. The latin word motus isn’t easy to translate, and “motion” often leads to misunderstandings. Because motus doesn’t necessarily mean local (physical) motion, but any transaction from potency to act, including the act of coming into existence. Therefore, if we say that God is the “first mover,” we do not mean that the world existed as some sort of clockwork, and that God “pushed” it into movement, but that God created it. We also need to explain what Aquinas meant by potentiality and actuality. Here I base myself upon Knut Erik Tranøy’s account of this teaching, as he explains it in his book Thomas av Aquino som moralfilosof (Thomas of Aquino as a moral philosopher.)
Potency and act is two prime forms or manners of existence. Tranøy explains that Aquinas says that a thing that is, is either in potentia or in actu. He explains further that to say that a thing is in potentia is to say that it has existence in a possible form; and to say that it is in actu is to say that it has existence in actuality/reality. And that it says that the possibilities (or rather potentialities) that a thing had when it was in potentia, is now — when it is in actu — actualized, realized, fulfilled.
The First Way is also backed up by the Second and Third Way in which he argues for the existence of “a first efficient cause” and “some being having of itself its own necessity.”
In the Second Way Aquinas claims that “[i]n the world of sense we find there is an order of efficient causes.” But most of these causes hinge upon other causes. Aquinas says that
in efficient causes it is not possible to go on to infinity, because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or only one. Now to take away the cause is to take away the effect. Therefore, if there be no first cause among efficient causes, there will be no ultimate, nor any intermediate cause. But if in efficient causes it is possible to go on to infinity, there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes; all of which is plainly false. Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God.
In the Third Way, which I find most interesting, and which is the argument most philosophically trained people regard as the best one, we find the argument from contingency. This argument, in my own words, basically says that every thing in existence have some sort of explanation for its existence, either in the necessity of its own being or in some external cause. But do we find anything in this universe that can explain its existence by “the necessity of its own being?” No, we don’t. And then the argument goes on, here in the words of Aquinas:
We find in nature things that are possible to be and not to be, since they are found to be generated, and to corrupt, and consequently, they are possible to be and not to be. But it is impossible for these always to exist, for that which is possible not to be at some time is not. Therefore, if everything is possible not to be, then at one time there could have been nothing in existence. Now if this were true, even now there would be nothing in existence, because that which does not exist only begins to exist by something already existing. Therefore, if at one time nothing was in existence, it would have been impossible for anything to have begun to exist; and thus even now nothing would be in existence — which is absurd. Therefore, not all beings are merely possible, but there must exist something the existence of which is necessary. But every necessary thing either has its necessity caused by another, or not. Now it is impossible to go on to infinity in necessary things which have their necessity caused by another, as has been already proved in regard to efficient causes. Therefore we cannot but postulate the existence of some being having of itself its own necessity, and not receiving it from another, but rather causing in others their necessity. This all men speak of as God.
Hope this helps!
Notes & references:
 Tranøy, Knut Erik, Thomas av Aquino som moralfilosof (Oslo: Universitetsforlaget/University Press, 1957), pp. 54-55. Here is the quote in its original norwegian:
Potens og akt er to grunnleggende værens-former eller værens-måter. Thomas sier om en ting som er, at den er in potentia eller in actu. Å si at en ting er in potentia er å si at den har væren i mulighets form; å si at en ting er in actu er å si at den har væren i virkelighet; og det vil igjen si at de muligheter tingen hadde da den var in potentia, nå — da den er in actu — er aktualisert, virkeliggjort, oppfylt.