It's cool that you are getting into science, and you are mostly right, but I think you are misunderstanding how cosmic expansion relates to us knowing the age of the universe.
Basically, there is no center of the universe. Sure, there is the center of mass, but god knows where that is, and it doesn't really matter. Because there is no center, for the sake of simplicity, lets take the earth as the center.
Back when the big bang happened, the clump of energy/particles/quantumness that would later become the earth was squished up against all the other energy/matter in the universe. Then BAM. Everything expanded.
Now, because we are centering ourselves on Earth, from our perspective, most all other matter is flying away from us at near the speed of light. Now, going to present day, we can look at this matter (what we now see as background radiation) and use red-shift plus the fact that we know it traveled away at near the speed of light to calculate that it has been moving away for about 13.8 billion years. So that's where we get the age of the universe from.
You are half right in saying there are galaxies beyond our 13.8 billion light year vision that we cannot see. We actually can see that far... sort of. But what we see at those ranges is what existed 13.8 billion years ago: nothing. So there are galaxies that are invisible to us, but that doesn't mean the universe is older or younger.