Final answer:
The Golgi apparatus is the cellular organelle that packages and directs proteins to their proper destinations, including the formation of lysosomes.
Step-by-step explanation:
The organelle responsible for packaging and directing proteins to their proper destination is the Golgi apparatus. It serves as the cell's 'mail room,' receiving proteins from the rough endoplasmic reticulum (RER). Once received, the Golgi apparatus organizes, modifies, packages, and tags these proteins with 'shipping addresses.' It can either ship these proteins to different areas within the cell or direct them outside the cell through the process known as exocytosis. The Golgi apparatus also plays a crucial role in producing organelles called lysosomes, which house enzymes meant for intracellular digestion.