The papers "Characteristic Classes and Homogeneous Spaces, I, II" by A. Borel and F. Hirzebruch is a classical resource. In general, homogeneous spaces form a rich class of examples to compute and play with.

It is useful to try to construct vector bundles with prescribed characteristic classes. The simplest example is that any class in $H^2(B)$
is the first Chern class of a complex line bundle over $B$, and any class in $H^1(B;\mathbb Z_2)$ is the first Stiefel-Whitney class of a real line bundle over $B$. This can be found in Husemoller's "Fiber bundles".

If you wish to have similar results for other vector bundles, you need to proceed by obstruction theory from maps from the base $B$ to the classifying space $BSO(n)$. Since the rational cohomology algebra of $BSO(n)$ is a polynomial algebra over appropriate Euler and Pontryagin classes, $BSO(n)$ is rationally a product $\Pi$ of Eilenberg-MacLane spaces with each factor corresponding to a characteristic class. Then we try to lift a map $B\to\Pi$ to $BSO(n)$. The conclusion is that after multiplying suitable classes in $H^*(B)$ by appropriate positive integers, one can realize them as Euler and Pontryagin classes of an oriented vector bundle over $B$. Also, these classes determine a vector bundle up to finite ambiguity (among all oriented vector bundles of a given fiber dimension). The only reference I know is to papers of mine (sorry!), see appendix B and appendix A even though this is standard material.