What really is an equation of state (EOS) and how are they developed?
They are merely brief and approximated ways to describe observed relations among state parameters. They do not need to have a theoretical background, but many times they have. It helps to create them and provides some reliability. The theory used to guide the construction of a EOS is anyone that turns to be useful.

how would he/she start and which script should he/she follow?

It is a very broad question. In thermodynamics there are few state parameters used normally. Thermodynamic theory is a macroscopic theory of somewhat general applicability, but EOS can be very specific, and this specificity can not be provided by thermodynamics, so, it turns to be a wide search.
In cases where simple general relations exist (mostly already discovered), it can be found just by fitting a data to a function.