Final answer:
The Golgi apparatus is the organelle responsible for packaging and transmitting cellular material, working in concert with the endoplasmic reticulum and Golgi vesicles within the endomembrane system.
Step-by-step explanation:
The organelle responsible for packaging and transmitting cellular material throughout the cell is the Golgi apparatus. The Golgi apparatus works closely with the endoplasmic reticulum (ER) as part of the endomembrane system. Proteins and lipids synthesized by the ER are transferred to the Golgi apparatus, where they're modified, sorted, and packaged into Golgi vesicles for transport either within the cell or for secretion outside the cell. Although ribosomes are involved in protein synthesis, and the ER contributes to protein processing and synthesis of lipids, only the Golgi apparatus has the specific function of packaging and transmitting these materials.