I'm 13 - I'd like to know how we found/discovered important details about the Universe. I know we have found many different things about the Universe, but when did we find how amazingly large the Universe is?
Throughout man's history, we have been learning things about the universe we live in.
Probably one of the most interesting discoveries that let us know how amazingly large the universe really is was not really very long ago, only in the 1920s when Edwin Hubble demonstrated that the Milky Way was not in fact the total sum of the universe, but just a galaxy, and there were other galaxies beyond our own, much further away than any of the stars that we see in the night sky with our naked eyes. This fundamentally changed our perspective of our place in the universe.
We have subsequently made many more significant discoveries about the nature of the universe, and many of those are covered in other questions already. If there are specific details you would like to know about, raise a specific question. The whole topic cannot be broached in one answer.
You're asking a very big question - basically, "what is the history of astronomy?" It's a very long story. Our knowledge about the size of the Universe grew slowly, bit by bit.
The ancient Greeks knew that Earth is round, like a sphere, and measured its size with surprisingly good precision.
In the 1600s, with the invention of the telescope, and the idea that the Earth revolves around the Sun, scientists (Johannes Kepler and others) figured out how measure the distance from Earth to Sun. After that, it was easy to measure distances within the Solar System.
In the 1800s, we made the next step: measure distances to other stars nearby. This is something called parallax.
In the 1910s, we figured out how to measure not just the distance to nearby stars, but the whole size of our galaxy. This was done by measuring variable stars (stars that go brighter, then dimmer, then brighter again, and so on).
After that, in the 1920s, Hubble figured how to measure the distance to other galaxies - that has been answered already on this page.
Different methods are used to measure different distances. Each method measures up to a certain distance; after that, a different method must be used. This is called the Cosmic Distance Ladder.
In any case, there's a lot to talk about on this topic. You may want to just grab a book on the history of astronomy and read. What you're asking is basically what's the whole history of astronomy, after all. :)
Here's a book that, even though pretty big, should be quite accessible to someone your age: