I'm looking for a comprehensive reference (for citation purposes) laying out the basic facts of the equivalence between $G$-spaces and bundles over $BG$ for a discrete group $G$. I'd like it to also include the identifications of the various auxiliary structures: $G$-representations to local systems, group cohomology to cohomology with local coefficients, (Borel) equivariant cohomology and cohomology of the total space of the bundle, the Hochschild-Serre spectral sequence and the Serre spectral sequence of the fibration.

This seems like foundational material but I've found this shockingly hard to find in a reference; can someone help me out?