Question

Which group of organelles is directly responsible for the production of new molecules within a cell?

Answer 1

New molecules are being made all over the place in any cell. That said,

Nucleus: production of RNA
Mitochondria: production of ATP
Ribosomes: production of proteins
Chloroplasts: production of simple sugars
or
Plenty of organelles make plenty of molecules, e.g. chloroplasts make carbohydrates and mitochondria make ATP.