Description of the residue field at a generic point
Solution 1:
Question: "How would you describe the residue field $\kappa(p)$ at a generic point $p\in\operatorname{Spec}(A)$ of a closed subset $V\subset\operatorname{Spec}(A)$?"
Answer: If $p \subseteq A$ is a prime ideal it follows the residue field $\kappa(p):=A_p/pA_p \cong K(A/p)$. Hence the residue field $\kappa(p)$ equals the quotient field $K(A/p)$ of the subscheme $Spec(A/p) $ defined by the prime ideal $p$.
Comment: I am guessing it to be the rational functions defined on $V$?
Answer: Yes, this is true.