1. ACROSS is used for the following two meanings:

A) from one side to the other side of something which has sides or limits such as a city, road or river
B) opposite side of something

2. THROUGH is used for movements from one side to another while surrounded by something; e.g.

a) Can you swim across this river? (means from one side to the another)
b) See those birds flying across the sky. (means from one side to the another)
c) Across the house, I can see some old friends. (means opposite side of the house)
d) She is waiting for me across the road. (means opposite side of the room)
e) The teacher drew a diagram showing how the blood flows through the heart.

INCORRECT: We walked across a number of villages yesterday.
CORRECT: We walked through a number of villages yesterday. (Of course the path on which we walked was surrounded by villages.)

INCORRECT: While going to Shimla our train passed across some tunnels.
CORRECT: While going to Shimla our train passed through some tunnels. (A tunnel is always surrounded by walls, etc.)

NOTE-I: When we move from one side to another ‘in something’ such as long grass or a forest, we use THROUGH (not ACROSS); e.g.

INCORRECT: I love walking across the forest.
CORRECT: I love walking through the forest.

NOTE-II: ACROSS/ON
When we talk about something extending or moving from one side to another, we use ACROSS, not ON.

INCORRECT: The papers were spread on the table.
CORRECT: The papers were spread across the table.

For more chapters/topics on English Grammar read the following book authored by me.

Link for buying the above book

For English Practice Sets on various topics read the following book authored by me.

Link for buying the above book