St. Thomas follows Aristotle in his solution of the regress problem: There must be an indemonstrable first principle because if everything were demonstrable, there would be an infinite regress; cf. his Expositio Posteriorum lib. 1 l. 7. Also, St. Thomas discusses in ibid. l.8 that circular demonstration ultimately leads to saying "if A is, A must be—a simple way of proving anything."
In Summa Theologica I-II q. 94 a. 2 c., St. Thomas states that
the first indemonstrable principle is that "the same thing cannot be affirmed and denied at the same time" [i.e., the principle of non-contradiction] which is based on the notion of "being" and "not-being": and on this principle all others are based, as is stated in Metaph. iv
primum principium indemonstrabile est quod non est simul affirmare et negare, quod fundatur supra rationem entis et non entis, et super hoc principio omnia alia fundantur, ut dicitur in IV Metaphys
Why is the principle of non-contradiction indemonstrable? Commenting on Aristotle's Metaphysics, St. Thomas writes (Sententia libri Metaphysicæ lib. 4 l. 6 [607.]):
[Aristotle] says, first, that certain men deem it fitting, i.e., they wish, to demonstrate this principle [of non-contradiction]; and they do this “through want of education,” i.e., through lack of learning or instruction. For there is want of education when a man does not know what to seek demonstration for and what not to; for not all things can be demonstrated. For if all things were demonstrable, then, since a thing is not demonstrated through itself but through something else, demonstrations would either be circular (although this cannot be true, because then the same thing would be both better known and less well known, as is clear in Book I of the Posterior Analytics), or they would have to proceed to infinity. But if there were an infinite regress in demonstrations, demonstration would be impossible, because the conclusion of any demonstration is made certain by reducing it to the first principle of demonstration. But this would not be the case if demonstration proceeded to infinity in an upward direction. It is clear, then, that not all things are demonstrable. And if some things are not demonstrable, these men cannot say that any principle is more indemonstrable than the above-mentioned one.
Dicit [Aristoteles] ergo primo, quod quidam dignum ducunt, sive volunt demonstrare praedictum principium. Et hoc propter apaedeusiam, idest ineruditionem sive indisciplinationem. Est enim ineruditio, quod homo nesciat quorum oportet quaerere demonstrationem, et quorum non: non enim possunt omnia demonstrari. Si enim omnia demonstrarentur, cum idem per seipsum non demonstretur, sed per aliud, oporteret esse circulum in demonstrationibus. Quod esse non potest: quia sic idem esset notius et minus notum, ut patet in primo posteriorum. Vel oporteret procedere in infinitum. Sed, si in infinitum procederetur, non esset demonstratio; quia quaelibet demonstrationis conclusio redditur certa per reductionem eius in primum demonstrationis principium: quod non esset si in infinitum demonstratio sursum procederet. Patet igitur, quod non sunt omnia demonstrabilia. Et si aliqua sunt non demonstrabilia, non possunt dicere quod aliquod principium sit magis indemonstrabile quam praedictum.
(This answer incorporates parts of this answer and this forum post; cf. this post.)
Courtesy GLeNotre's answer, here's: