In this work we approach the theory of finitely generated FI_G-modules through the language of abelian categories. In the first chapter, we define various homological invariants of finitely generated FI_G-modules, and show that they encode certain previously observed phenomena. More specifically, we show that the derived functors of the derivative encode the so-called Nagpal number of the module. We also provide a theory of depth for finitely generated FI_G-modules, which generalizes previous work of Sam and Snowden. In the second chapter, joint with Liping Li, we study a theory of local cohomology for FI_G-modules. Using this theory we refine the results of the first chapter, while expanding it in many ways. It is shown that these local cohomology modules encode a plethora of significant properties, including the regularity of the module. Finally, the third chapter deals with removing the assumption of finite generation from the first two. We prove that the weaker assumption of coherence is sufficient for much of the theory to continue to work. As an application, we prove a kind of local duality for coherent FI_G-modules.