Across and over can both be used to mean to the other side of a line such as a river, road, bridge, etc.
Across is used to mean on the other side of a flat area or surface.
Over is used to mean on the other side of something high.
Through is used for movement in a three-dimensional (3D) space.
- His house is across/over the river.
- He walked across the desert.
- The thief climbed over the wall.
- They went for a walk through the forest.