How do the different organelles in a eukaryotic cell work together to produce a hormone such as insulin?

Hormones are proteins. Each specific protein is made up of a specific sequence of amino acids. DNA in the nucleus codes for the specific sequence. This DNA is transcribed into mRNA which leaves the nucleus through the nuclear pore and is translated by ribosomes into the specific sequence of amino acids (i.e. protein is produced). The protein then travels via the rough endoplasmic reticulum to the Golgi apparatus, where it is modified and packaged into vesicles. The vesicle containing the protein moves via the cytoskeleton towards the cell surface membrane and fuses with it. This allows the hormone (protein) to be released out of the cell by a process called exocytosis.