I am reading A Course in Arithmetic by J-P Serre. Definition $7$ on page $32$ says
Two quadratic forms $f$ and $f'$ are called equivalent if the corresponding modules are isomophic.
I am not sure what modules Serre is referring to. What would be the overlying set, which is made up of these equivalence classes?
I'm not sure I understand your question. A quadratic form $Q$ is always defined on an $A$-module $V$, where $A$ is a commutative ring (cf. p. 27). A quadratic module (also known as a quadratic space) is just a pair $(V,Q)$, where $V$ is a module and $Q$ is a quadratic form on $V$; this definition is also given on p. 27.
In this definition, Serre simply defines two quadratic forms $Q, Q'$ on $A$-modules $V, V'$ to be equivalent if the quadratic modules $(V,Q)$ and $(V',Q')$ are isomorphic. (A morphism of quadratic modules is an $A$-linear map $f: (V,Q) \to (V', Q')$ such that $Q' \circ f = Q$; this is also defined on p. 27.)