Through
is used to talk about movement in an area which is accepted to be 3-dimensional (it has height, width and length) examples are:
a room
a doorway
a forest
a school
a housing estate
the sea (including depth)
Across
is used to talk about movement over a surface that is considered to be 2-dimensional (it only has length and width) examples are:
a table
a bed
a park (not considering trees as important)
a piece of paper
the surface of a lake
Along
is used when we consider movement in a linear direction, we are only concerned with its length, examples are:
a path
a road
a tightrope