ag.algebraic geometry - Tautological bundle on G$(n,k)$ and Chern classes

Depending on what you need to do with the Chern roots, it may be cleaner to just ask for the Chern classes of $S$.

In this case, let $Q$ be the quotient bundle, i.e., there is a trivial bundle ${bf C}^k$ which contains $S$ as a subbundle, and $Q = {bf C}^k / S$. The $i$th Chern class of $Q$ is the cohomology class of $sigma_i$, the special Schubert class of codimension $i$ (see Proposition 3.5.5 of Manivel's book Symmetric Functions, Schubert Polynomials, and Degeneracy Loci). Using the relation $c(S)c(Q) = 1$, and knowledge of the cohomology ring of $G(n,k)$ should be enough to perform any usual calculations.

For computing with these Schubert classes, one only needs to learn the Pieri rule (and perhaps the Littlewood-Richardson rule depending on the circumstance), both of which can be found in Manivel's book (chapter 1) or see Wikipedia.