Into what organ is pepsin secreted to begin the chemical digestion of proteins?

Small intestine.

Small intestine, specifically the duodenum.

It is the stomach, but it's not really pepsin that is secreted from the glands of the stomach lining. It is its inactive form, pepsinogen, that is secreted into the lumen of the stomach. The pepsinogen is then activated by the acidic pH of the stomach and it is converted to pepsin, which is the active enzyme that begins digesting proteins. HCl (hydrochloric acid) is secreted into the stomach from specialized cells in the stomach lining called zymogen cells. The reason that pepsin is not made in its active form within the cells of the stomach lining is that if it were, it would begin to digest the stomach itself, which is made of smooth muscle proteins. The stomach is protected from this by the secretion of a thick mucus lining, also produced by mucosal cells in the stomach wall. It's vital that the pepsin does not become active until it is mixing with the food in the stomach, not in the walls!