At the tip of an actively growing root is the root cap. The root cap is a loose assemblage of cells and a slimly substance called mucilage that protects and lubricates the root tip as it forces its way through the soil. Behind the root cap is the zone of active cell division were new cells are created. Immediately behind is the zone of cell elongation. This is where the new cells begin growth.
Plant cells grow by pumping themselves full of water.
They do this by actively moving salt ions, usually potassium, into the cell. They also move sugars into the cell. The name of the game is to be saltier and sweeter that the water around you. This causes water to move into the cell. The cells do not actively take up water.