Across is used to show movement from one side of a surface to another. As long it has edges or limits, both the size of the surface (and the means of reaching the other side of it) is irrelevant in the usage of across. You could be walking across a room or flying a spaceship across the galaxy and it wouldn’t make a difference, as long as there is another side that can be reached.

To help with understanding how to apply this rule, imagine that you are standing on one side of a river, and you want to reach the other side. That means you want to go across it. Beside you is a small rowboat and a short distance away is a bridge. So, how can you achieve your aim? Well, you can swim across the river, get in the boat and row across it, or make your way to the bridge and walk across it.

With this preposition, it is important to note that you do not need to be in contact with a surface to be able to move across it. For example, if you got on a flight from Ireland to America, you would fly across the Atlantic Ocean without ever touching it.  

Examples:

“I walked across the road.”

“He drove across Italy.”

“I have never flown across the Pacific Ocean.”

“She drew a line across the paper."

Through is used to describe moving into the middle of something three-dimensional and then out the other side of it, such as driving through a tunnel, or walking through a door. The need to move or pass through something to reach another side is the main point that separates 'through' from the similar preposition 'across'.

Take a forest, for example. A forest is covered by trees, so when you enter one you are enclosed within the lines of trees and you have to pass through them to reach the other side. Therefore, we say, “I walked through the forest”, rather than, “I walked across the forest”.

Grass can be used as another example. Take a look at these two pictures:

In the first picture, a cat is walking while enclosed within thick, tall and overgrown grass, like the kind you might find in a field in the countryside. The correct way to describe that would be to say, "The cat is walking through the grass", as it has to physically move through it rather than simply walk across it. In the second picture, the cat is no longer enclosed as the grass is much shorter, like a recently mowed lawn, and so the correct way to describe it would be to say, “The cat is walking across the grass.”