https://en.wikipedia.org/wiki/Five_Ways_(Aquinas)#cite_ref-IEP_aris-mot_14-0

Aquinas argument from change is presented as follows:
"In the world, we can see that at least some things are changing. Whatever is changing is being changed by something else. If that by which it is changing is itself changed, then it too is being changed by something else. But this chain cannot be infinitely long, so there must be something that causes change without itself changing. This everyone understands to be God.[7][13]"

The word "motion" has a technical definition:
"Aquinas uses the term "motion" in his argument, but by this he understands any kind of "change", more specifically a transit from potentiality to actuality.[14] Since a potential does not yet exist, it cannot cause itself to exist and can therefore only be brought into existence by something already existing.[1]"

The kind of infinite causation that Aquinas rejects is an infinite number of changers that need to exist for our actual state of affairs to exist:
"When Aquinas argues that a causal chain cannot be infinitely long, he does not have in mind a chain where each element is a prior event that causes the next event; in other words, he is not arguing for a first event in a sequence. Rather, his argument is that a chain of concurrent or simultaneous effects must be rooted ultimately in a cause capable of generating these effects, and hence for a cause that is first in the hierarchical sense, not the temporal sense.[8]"

"Aquinas follows the distinction found in Aristotle's Physics 8.5, and developed by Simplicius, Maimonides, and Avicenna that a causal chain may be either accidental (Socrates' father caused Socrates, Socrates' grandfather caused Socrates' father, but Socrates' grandfather only accidentally caused Socrates) or essential (a stick is moving a stone, because a hand is simultaneously moving the stick, and thus transitively the hand is moving the stone.)[9]"

"An accidental series of causes is one in which the earlier causes need no longer exist in order for the series to continue. ... An essential series of causes is one in which the first, and every intermediate member of the series, must continue to exist in order for the causal series to continue as such.[1]
— "Agellius" (paraphrasing Fesser), The First Cause Argument Misunderstood"

"Aquinas expands the first of these – God as the "unmoved mover" – in his Summa Contra Gentiles.[1]"

It looks like this argument goes as follows:
Premise 1. All things that change are caused to change by something external to the changing thing.
Premise 2. Everything in the universe is changing.
Conclusion. Therefore, everything in the universe is caused to change by something external to the universe.

From the quotes above, there are two kinds of causation- accidental and essential.
The kind of causation relevant to the above argument is "essential" causation and Aquinas thinks that such causation is not infinite while allowing for the possibility of infinite accidental causation where the past is without beginning. Thus, the reasoning goes that the universe changes only if something external to the universe causes the universe to change. If that external thing stopped changing the universe, then everything in the universe would not change. Looking back at the argument, I think most people will accept premise 2 (perhaps eternalists won't?) and, for sake of argument, let us accept that everything in the universe is changing.

Possible objection to premises:
I think premise 1 may bother some people. They may ask, why believe that whatever thing that changed was caused to change instead of believing that change is a fundamental feature of reality?

How could something that is unchanging (and thus timeless) cause every change in the universe:It looks like the argument assumes that change can only be accounted for by something that is not changing, but that something is somehow the cause of every other changing thing.
This part of the argument confuses me. How could a changeless thing, cause changes to occur and remain changeless? How could God's act be changeless, simultaneous to every change in the universe and be the cause for all changes in the universe and still remain changeless?

God's act of changing the universe vs. every other change that occurs:
Consider the example of the hand that uses a stick to change a rock’s location, the hand is simultaneously moving a stone and without the hand the stone would not move. Thus, this is an essential series of changers. Aquinas wants to argue that without God’s act of changing the universe, the universe would not change at all.

“An essential series of causes is one in which the first, and every intermediate member of the series, must continue to exist in order for the causal series to continue as such.”

The hand, stick and rock example does not serve as an example of an essential sequence. The event of the hand moving the stick that moves the rock are events that occur in a temporal sequence, but I think that that really does not matter so long as we can give “essential ordered causation” a definition:

Essential ordered causation: Y exists and changes only if X causes Y to exist and changes Y.

The kinds of changes that occur in the universe ( Y ) are unlike the changes that God ( X ) is responsible for. God somehow changelessly changes the universe, but everything in the universe changes everything else in the universe and does so by experiencing changes themselves.

William Lane Craig's criticism of the essentially ordered causes:
"The principal argument used to eliminate such a regress is that in essentially ordered infinite regress of causes, only instrumental causes would exist, and, hence there would be no intrinsic causality in the series to produce the observed effect. The defender of this argument faces, however, this dilemma: if an instrumental cause is defined as a cause lacking intrinsic causal efficacy, one cannot preclude an infinite regress of instrumental causes each receiving its casual efficacy extrinsically from its predecessor but if an instrumental cause is defined as a cause depending ultimately upon a first cause, then it cannot be shown that the causes in an infinite regress are truly instrumental. Even should this dilemma be irresolvable, however, it could still be the case that an infinite regress of essentially ordered causes is intuitively implausible: it would mean, for example, that a watch could run without a spring if it had an infinite number of gears or that a train could move without an engine simply by having an infinite number of box cars." (289, The cosmological argument from Plato to Leibniz).

Joseph M. Magee's criticism of the First Way:
"... premise (II) is not true: not all motion or change (even purely physical motion) is caused by something other than what is changing. Rather, some things change of themselves through the exercise of intrinsic physical forces."
http://www.aquinasonline.com/Topics/firstway-assess.pdf