Short version:
There are very good reasons to reject the first premise, which takes the third premise with it. The second premise is plausible and generally accepted, but not without problems of its own. All of this ultimately irrelevant, however, because the argument is invalid. The conclusion does not follow from the premises even if we accept them all as true.
Long version:
As a bit of philosophical history, it’s worth noting that this argument was originally the second part of Aquinas’ Five Ways argument (in which he presents five arguments in favor of the existence of God). In all likelihood, Aquinas himself did not think that this argument proved anything on its own. The most plausible and charitable interpretations of Aquinas of which I am aware propose that he thought it was only when all five arguments were taken together that they formed a proof of God’s existence.
As already noted by @gorillapaws, the Second Way is known as the cosmological argument when taken as a separate entity and is distinct from the teleological and cosmological arguments. The cosmological argument posits God as the only possible explanation for why there is a universe at all, whereas the teleological argument says that God is the best explanation for the apparent order we see in the universe and the ontological argument says that the existence of God is logically necessary.
As for the cosmological argument itself, then, there are many problems (some of which have already been noted).
1. The universe is not infinite—it began to exist.
First, we need to clarify what this premise means. The claim here is that the universe has not existed for an infinite amount of time. It is not a claim about the universe’s size. So the premise could be usefully restated as “the universe has not always existed—at some point it came into being.”
This, as it turns out, might not be true. At the very least, we need not accept it as a given. While the Big Bang is the prevailing model of cosmology in science, that theory does not necessarily entail that the universe began to exist in the sense needed for this premise to be true. As such, the premise is open to rejection.
2. Everything that begins to exist has a cause.
This premise is most likely true, and I think most people would accept it outside of an argument for the existence of God. If something began, it stands to reason that it had conditions for beginning and not already existing. There are hypotheses of quantum mechanics that contradict the claim, however, so maybe it will turn out not to be true (as surprising as that might be). But the mere possibility that it might be false isn’t really grounds for rejecting the premise. As such, we might at least grant this premise for the sake of argument (pending further investigation).
3. The universe had a cause.
This premise is a sub-conclusion and rests on our accepting both the first and second premises. Even if we accept the second premise for one reason or another, then, the reasons for rejecting the first premise are sufficient for rejecting the third as well.
4. The cause had to be something that always existed, i.e. God.
Perhaps the biggest problem with the argument is that this conclusion simply does not follow from the premises. Assume that all three premises are true: the universe would have to have a cause, but that cause could itself be something that came into existence (cf. theories that hypothesize that universes are born from the black holes of other universes).
In Aquinas’ original argument, this objection is parried by an argument about the (putative) impossibility of infinite regresses. The black hole theory was not around in Aquinas’ time, of course, but that’s the mark of a great philosopher—thinking of replies to objections before the objections even exist. Given his argument about infinite regresses, then, Aquinas could argue as follows: even if our universe was born from the black hole of another universe, that universe had to come from somewhere until eventually something had to be created outright.
There are reasons for doubting that conclusion, but we can leave them all aside. As before, let us assume that all three premises are true. Let us further assume that the infinite regress argument is true. At best, all this would mean is that the universe was caused by something that has always existed. It does not prove that this thing is God. Eternal existence might be a necessary condition for God’s existence (that is, anything that does not exist eternally does not count as God), it is not a sufficient condition for God’s existence (that is, it takes more than merely existing eternally to be God; said another way, God has more properties than eternal existence).
Consider: would any of the Abrahamic religions consider their views to be vindicated if it could be proven that the universe had a cause that existed eternally but was not omniscient, omnipotent, and omnibenevolent? The answer is clearly “no.” It takes more to prove the existence of God than proving the existence of an eternally existing cause of the universe. Thus the argument is defeated on two levels: first, there is reason to reject at least two of the premises (and maybe all three); second, the conclusion would not follow from the premises even if were to accept all three premises. Therefore, the cosmological argument—or at least, this version of it—does not prove the existence of God.
It is important to remember, however, that refuting an argument is not the same as refuting it’s conclusion. God could still exist even if the cosmological argument fails to prove His existence.