What really is an equation of state (EOS) and how are they developed? Are EOSs merely correlations that mathematically represent (in a better or worse way) experimental data or do they have a theoretical background? How are they developed (suppose one wishes to develop his/her own EOS -- how would he/she start and which script should he/she follow?).
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.