If the two premises are true then the conclusion logically follows. However neither premise is necessarily true. Take the first premise "Whatever begins to exist has a cause." The sun's gravity causes the earth to orbit the sun. Newton's first Law of Motion states that "an object continues in a state of rest or uniform motion in a straight line unless acted upon by an external force." Gravity is that external force. But the earth does not actually orbit the sun, rather it orbits the center of mass of the sun-earth system. The earth's gravity causes the sun to orbit that same center of mass, albeit the radius of the sun's orbit is much smaller. This is Newton's third Law of Motion "for every action there is an equal and opposite reaction." The sun acts upon the earth and the earth acts upon the sun. Now is there any reason the think that the sun acted upon the earth before the earth acted upon the earth. I can see nothing that would force that conclusion. I am forced to conclude that the two causes occurred simultaneously. Quantum entanglement suggests that this is highly probable.

So if something caused something else to begin to exist then that something else must also have caused the something to begin to exist.

As for the second premise "The universe began to exist" there is no evidence that this is true. Theists will point to the Big Bang (possibly simultaneously claiming that the universe is 6,022 years old) but the Big Bang was not necessarily the beginning of the universe. For one thing the Big Bang may have been preceded by a Big Crunch. Then there is the Gamow model of the oscillating universe where the is a Big Bang, then a period of expansion followed by a period of contraction and a Big Crunch, repeated forever. The discovery of dark energy seems to have created a problem for the Gamow model but I don't think cosmologists have given up on it entirely.

Finally there is the Thomist non-sequitur that the first cause is understood to be God. It just ain't so (apologies to Gershwin).

According to conjectures of quantum gravity loop theory, from a cosmic perspective there’s no such thing as time. What we experience as time and space are only modes of thought peculiar to our way of interpreting reality, and they have no true essence. Clearly we don’t understand the meaning of existence, because in our human sense-realm things exist through time. If there is no time it is meaningless to even talk about creation. Also meaningless are position, distance, motion, speed, etc. There are no objective “things” either. Particles of matter are not things but interactions.

In that we don’t know what the hell we’re talking about it seems smart not to argue over the existence of God. “God” is a label for stuff we don’t know and can’t understand.

Since I believe there’s a label called “God” that designates what is unknown. Does that make me a theist?

My first objection is that the word “God” hasn’t been defined.
My second objection is that the second premise, that the universe had a beginning, is an unproven assumption.
My third objection is that, if the universe had a cause, it is no indication that such cause must have been something best described as a God.