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.
http://en.wikipedia.org/wiki/Eratosthenes
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.
http://www.tbp.org/pubs/Features/Su04Bell.pdf
In the 1800s, we made the next step: measure distances to other stars nearby. This is something called parallax.
http://en.wikipedia.org/wiki/Stellar_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).
http://www.aip.org/history/cosmology/ideas/island.htm
After that, in the 1920s, Hubble figured how to measure the distance to other galaxies - that has been answered already on this page.
http://en.wikipedia.org/wiki/Edwin_Hubble#The_universe_goes_beyond_the_Milky_Way_galaxy
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.
http://en.wikipedia.org/wiki/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. :)
http://en.wikipedia.org/wiki/History_of_astronomy
Here's a book that, even though pretty big, should be quite accessible to someone your age:
http://www.amazon.com/The-History-Astronomy-Heather-Couper/dp/1554075378/
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