Am I right in thinking that the human cell, getting instructions from the DNA .....?

Yes. The genome (meaning the collection of all the cells' DNA) contains "instructions" on how to create every type of cell in the body. Cell types are determined by the patterns of which genes are turned off and which are turned on in different patterns. This in turn is controlled by incredibly complex networks of regulatory molecules, which all starts being established a few seconds after the sperm fertilizes the egg.