By Swan's Theorem, we know that projective modules over a ring are an algebraic analogue of vector bundles over a base space.
Is there some sort of cohomology theory of rings (or modules? or schemes, maybe?) such that, given a projective module over $R$, there are natural choices of classes in the cohomology of $R$, the same way that vector bundles over $B$ have characteristic classes in the cohomology of $B$?
Perhaps this requires an analogous notion of a classifying space...
Any insight (or maybe an explanation as to why this wouldn't be possible) would be appreciated.
There are Chern classes defined on K-theory groups with values in cyclic homology and Hochschild homology, which are the exact analogue of characteristic classes for general algebras —in fact, these can be defined using those.
You can find this explained in Loday's book on cyclic homology, in Karoubi's book on the same subject, in Rosenberg's introduction to K-theory, etc.