Books on topology and geometry of Grassmannians

I enjoyed the book by Milnor and Stasheff, "Characteristic Classes." This explains the business of the universal bundle, and the cohomology ring (which is to say, characteristic classes).

As for the algebraic case...this is explained in the new edition of EGA I, but it is a little technical (relying on a standard lemma in basic moduli theory, which I found rather difficult to understand). There are also explanations in the book "FGA Explained."

This is an answer to your continued question on MO. https://mathoverflow.net/questions/73736/topology-and-geometry-of-grassmannians-g-k-mathbbrn-or-g-k-mathbbcn

(1) What would you want out of a "topological classification"?

(2) Yes, Grassmannians can have exotic smooth structures. For example $\mathbb RP^n = G_1(\mathbb R^{n+1})$ has well-known exotic smooth structures for various $n$.