The unmoved mover argument for God goes something like this. Nothing moves without first being moved by something else. However, there is motion, so there must also be an unmoved mover. That mover is God.
I think this argument gets you somewhere, but not quite to God.
One argument against it is that there is no beginning to motion. Motion just goes on infinitely backwards. Possible, although evidence for the big bang undermines this argument.
Another counter-argument is that, while there is a first mover, that mover is not God. There is simply some other cause of motion. The rejoinder is that the unmoved mover is by definition God, regardless of what you call it.
But I don't think that saying the unmoved mover is, by definition, God is quite right. Sure, we can call the unmoved mover God, but that's not really the God that we're trying to prove-- a God that takes some interest in our lives. Proving that there is a thing or phenomenon that causes motion without being moved itself does not prove that that thing cares about us humans.
But I do think the existence of motion creates some problems for atheists. Assuming motion does not go back infinitely as the big bang suggests, science currently cannot explain motion. God's existence, however, can explain motion.
Atheists might argue there is a non-god unmoved mover, we just don't know what it is. But without some evidence, this is just a declaration of the atheist's faith. Believing in a non-god explanation for motion is no more based on evidence and reason than believing in God.