Bunke's notes are indeed a great source for this material! However, in order to get to the main definitions and properties, he does breeze through a lot of the fundamental prerequisites. I will briefly add to the great list Omar has provided above.

For a classical, non sheaf theoretic approach: Jeff Cheeger, James Simons, *Differential characters and geometric invariants*. http://numr.wdfiles.com/local--files/differential-cohomology/cheeger-simons.pdf

For a great text which models differential cohomology via Deligne cohomology (and discusses the relationship with $U(1)$-gerbes in low degrees): Jean-Luc Brylinski, *loop spaces characteristic classes and geometric quantization*.

If you have a strong background in category theory (in particular categories of sheaves), I would strongly recommend the higher stack model of differential cohomology (similar to Brylinski's gerbes, but generalized to higher dimensions) which is discussed in: Urs Schreiber, *Differential cohomology in a cohesive infinity-topos* https://arxiv.org/pdf/1310.7930.pdf

For the original definition in terms of differnetial function spectra, you can see: Mike Hopkins, Isadore Singer, *Quadratic Functions in Geometry, Topology,and M-Theory* https://arxiv.org/abs/math/0211216

If you want to see a brief summary of Bunke's model for differential cohomology theories (the papers that Omar mentioned are far more comprehensive and thorough, but just to add a bit more to the list), there is also a short discussion in this pre-print of mine with Hisham Sati: *Spectral sequences in smooth generalized cohomology* https://arxiv.org/pdf/1605.03444.pdf