A potential argument for existence of God

[Esquilax]

Anything that moves must rationally have a beginning,

Therefore, your god needs a beginning... or doesn't he move? Either that, or you believe that things can move without beginnings. Either way, you're screwed at the first hurdle.

because if there was an infinite amount of time before a certain movement, then that movement would never come to be (because you would have to go through an infinite number of years before you 'reached' the movement). So moving objects (including our universe) couldn't have existed forever. They must have began.

I don't think you understand how time works. It's pretty standard for theists, taking simple things and turning them into impossible challenges, reliant on magic, if they think it'll prove their god. But if you have an infinite amount of time, by definition you have enough time to reach any given event along a timeline, regardless of how lengthy that timeline might be. There are infinite numbers, but it's not impossible to count to four.

Apart from this, most empirical evidence suggests that the universe had a beginning, such as the Second Law of Thermodynamics, expansion of the universe, regularity of cosmic microwave background, and data from the BICEP2 (including direct evidence of gravitational waves). Most physicists and cosmologists agree that the universe did have a beginning. As Dr. Pluijm of the Universtiy of Michigan said, "The scientific evidence is now overwhelming that the Universe began with a 'Big Bang' about 15 billion years ago. The Big Bang theory is the most widely accepted theory of the creation of the Universe." Dr. Louis Clavelli, professor of physics at the University of Alabama, similarly reaffirms: "A large body of astrophysical observations now clearly points to a beginning for our universe about 15 billion years ago in a cataclysmic outpouring of elementary particles." Stephen Hawkings, after giving a lecture on time, said: "The conclusion of this lecture is that the universe has not existed forever. Rather, the universe, and time itself, had a beginning in the Big Bang, about 15 billion years ago."

The big bang does not necessarily connote the very beginning of the universe, just the beginning of its current, expansionary state. In fact, if you would look further at the science you've paid lip service to, right up until the moment that it seems to confirm what you already believe and not a word further, you would see that the common consensus offers no view of what the universe was like before the Planck time, as we do not have the technology, nor knowledge, to predict that with any degree of accuracy. In fact, given that time and space commingle, the point before the big bang, where space itself was in a state unlike anything we've ever observed, could easily also be a point at which time behaves differently, such that your "everything that begins to exist has a cause," crap is no longer even applicable. Your self serving assumption that time has behaved exactly the same way, even before the point at which it even could, simply does not stand.