Classical Lie superalgebras arise in the physical theory of supersymmetry and behave analogously to Lie algebras in the theory of algebraic groups. In the theory of Lie algebras, the Koszul resolution is finite, meaning relative cohomology rings are finite as a vector space over the field. For Lie superalgebras, the Koszul resolution is infinite. A theorem of Boe-Kujawa-Nakano states that for classical Lie superalgebras, the cohomology ring relative to the even component has finite Krull dimension, and the structure of this cohomology ring is determined by the invariant theory of a reductive group's action on a vector space. This realization opens the door to the study of the support variety theory for classical Lie superalgebras. The main result of this thesis is a generalization of the result of Boe-Kujawa-Nakano. The main theorem asserts that the cohomology ring of a classical Lie superalgebra relative to any even subsuperalgebra has finite Krull dimension, and is indeed a finite extension of a subquotient of the Boe-Kujawa-Nakano cohomology ring via restriction. The proof of the main theorem relies on a spectral sequence inspired by that of Hochschild-Serre. The spectral sequence used to prove finite generation proves to be an invaluable tool in analyzing the behavior of cohomology rings. An example is presented in which the Krull dimension of a relative cohomology ring is positive but not equal to the Krull dimension of Boe-Kujawa-Nakano cohomology. Conditions are given for when the cohomology ring will be Cohen-Macaulay. With finite-generation established, the final chapter of this dissertation is devoted to studying the relative support variety theory for modules. A realization morphism induced by restriction of functions plays a role similar to that of Friedlander-Parshall's realization morphism for restricted Lie algebras. The main goal of this chapter is to work towards a conjectural tensor product theorem for Lie superalgebras, which would generalize results of Grantcharov-Grantcharov-Nakano-Wu. To this end, rank varieties are introduced which conjecturally generalize the rank varieties of Grantcharov-Grantcharov-Nakano-Wu.