To understand Aquinas, we must understand his words.
The first and more manifest way is the argument from motion. 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.
--- Aquinas Understanding ---
I'll explain this to show the disparity in its meaning and Dawkin's understanding thereof.
You must understand actuality and potentiality. Read this: http://www.newadvent.org/cathen/01124a.htm
All substances actually exist, and their future existence is not actual but potential.
Example: I have a ripe apple. It is actually ripe. It potentially rots, or potentially stays ripe. With change (the movement from potentially to actuality) it either stays ripe or rots.
Now, nothing in motion is prior to itself. That is, nothing that exists potentially can actually move itself into actuality. Example: All the apples that potentially exist from now to 2023 cannot be the source of their future existence. Something that actually exists must move them, that is, something that actually exists must move from the potentiality of causing those potential applies into actuality.
Therefore, as the world around us displays a ton of things in change, it is apparent that there must a non-moving (from potence to actuality), universal substance, else nothing would move (from potence to actuality). That is the basis of Aquinas' God. Aquinas observed that the existence of things in motion necessitates something not in motion which sets all the others in motion; something whose unchanging essence is existence itself.
--- Dawkins' Understanding ---
Taking @stoicfury's interpretation:
Stated differently, if infinite regresses are part of the way things are, and God exists, he too would be subject to infinite regress (otherwise infinite regresses aren't part of the way things are). Where did he (God) come from? Where did the thing that created God come from? Where did the thing that created the thing that God came from come from? etc. etc. etc. It doesn't make sense to invoke infinite regresses and simultaneously invoke something that makes them very not infinite.
This shows fundamental misunderstanding. Things in motion (from potence to actuality) must be moved, yes. But God does not move from potence to actuality. He IS (for He does not change, He is existence itself wherefore all things derive their being. Put another way, He is the Sustaining Principle.)
Others claim that Aquinas must justify why there's an exception. However, no exception exists. They misunderstand him. Aquinas' proof does not include infinite regress. Their misunderstanding of him does.
--- Conclusion ---
Dawkins does great at misinterpreting Aquinas' arguments and tearing his mistaken understandings thereof apart. But does Dawkins critique Aquinas? No, Dawkins critiques Dawkins' perception of Aquinas.
To understand Aquinas, don't read "proofs" of his "fallacies" by those who misunderstood him. Read Aristotle, to get an idea of actuality and potentiality, whose method and foundation Aquinas builds upon. Finally, read Aquinas.