'Across' refers to moving from one side to another.
Ex - Ram travelled across India on his motorcycle. / Ram and Shyam are swimming across the lake.

'Through' refers to moving directly inside something and out the other end.
Ex - The bullet shot went through the window. / The train passes through the tunnel.

'Into' refers to entering or looking inside something.
Ex - James went into the room. / They stare into the darkness.