The cellular organelle responsible for packaging and transporting proteins is the Golgi apparatus. This organelle modifies proteins synthesized in the rough endoplasmic reticulum and prepares them for distribution within or outside the cell. The Golgi apparatus processes these proteins by adding carbohydrates or lipids, forming glycoproteins or glycolipids. Once modified, proteins are sorted and packaged into vesicles, which transport them to their designated locations, ensuring proper cellular function and communication.