Aquinas' argument is a "cosmological" one - that is, he argues that we can know that God exists by using our knowledge and experience of the physical world. However, it seems to me that the conclusions he reaches are more matters of opinion than "proofs".

For example (at the risk of oversimplifying), the first proof is:

Everthing that we observe which moves has its movement caused by another moving body, therefore there must be a something capable of creating movement independently which started the whole thing off.

The "therefore" clause is what presents a problem. A reasonable counter-"proof" might be:

Everything that we observe which moves has its movement caused by another moving body, therefore nothing can exist which can impart movement unilaterally

We can have no way, based purely on our experience of the physical world, of "proving" either of these statements. I think that the first four proofs all fall into this category.

I think there is a fundamental problem with the "cosmological" approach: it means you are attempting to prove the existence of a transcendental God who operates outside the boundaries of our physical world, but doing so based on premises whose scope is restricted to the physical world. This is like trying to paint a rainbow when all you have is red paint, or to build a bridge when all you have is a pencil and paper: the raw materials you have to work with are not capable of accomplishing the task.

The fifth proof, which appeals to the "design" is one where Aquinas probably has in mind the natural world, and as such he might seem particularly vulnerable, in the light of evolutionary science. However, recent scientific research does seems to point to there being some kind of order in the universe - Martin Rees' book "Just Six Numbers" describes how the universe could not exist if any of six constants (such as the force of gravity) had slightly different values. You might argue that the numbers might well have been different, in which case we wouldn't be here to worry about it. Rees quotes an argument from the French philosopher Leslie:

Suppose you are facing a firing squad. Fifty marksmen take aim, but they all miss. If they hadn't all missed, you wouldn't have survived to ponder the matter. But you wouldn't just leave it at that - you'd still be baffled, and would seek some further reason for your good fortune.