The Big Bang theory is probably the best place to start. You see, in astronomy an observatory may detect that a star is moving away from us (the Earth). The same telescope (or another) might lok in the opposite direction and see that a star over there is miving away from us as well. These are astronomical observations; such and such a star moving at such and such a speed.
When many of these observations are put together to form a single coherent picture of the universe, i.e., "all of the stars are moving away from us, and the more distant ones are moving more rapidly", that is cosmology. From this observation, it was concluded that in the past, all the stars were closer together. The further back in the past we look, the the closer together the stars would be. Therefore, if we look far enough back, all the stars were together in one place. This is the big bang theory, and it is a cosmological model.
You talked about the speed of expantion and that, the farther stars expand with a higher speed. can you tell me about the relativity and the limit of light's speed???