Across and over can both mean ‘on or to the other side of a line, river, road, bridge etc’.
You can use both in this way:
His village is just across the border.
He ran across the finishing line and raised his arms in victory.
But here’s the difference that you need to keep in mind:
Over is used to mean on or to the other side of something high.
The cat jumped over the fence and disappeared.
Across is used to mean movement from one side to the other side of a flat area or surface.
The river is too wide. We can’t swim across.
Use through for a movement in a three-dimensional space, with things on all sides.
We drove through the town in search of an antique shop.
We drove across the desert.
Do you see the difference?