While the Standard Model is an achievement of our understanding of the physical world down to fundamental levels, it is incomplete. Namely it cannot address the Hierarchy between the Higgs mass and the Planck scale, or provide a particle description of what constitutes dark matter. In this thesis I review these two issues and then outline my novel research to motivate models of dark sectors as potential solutions, and how we can test these theories with astrophysical probes. First, I develop tools for modeling hadronization in confining dark sectors without light dark quarks, allowing the showering of dark sector glueballs to be simulated with uncertainty estimates. I then use this work to constrain models of dark matter that annihilate to dark glueballs with indirect detection data from Fermi-LAT and AMS-02. I also present work updating this model of hadronization, and show how these dark sectors could be studied at the LHC. Lastly, I consider another dark sector, atomic dark matter, and present work studying its behaviour on subgalactic scales using cosmological simulations