I wonder if real Grassmannian manifold $SO(p+q)/SO(p) \times SO(q)$ have nontrivial first Pontryagin class? I only have physics background and know really little about characteristic class theory.
This is a partial answer:
If $p=1$ or $q=1$ then this manifold is the $n$-sphere which, therefore, has zero Pontryagin classes.
If $p=2$ or $q=2$, then this manifold always has $p_1\ne 0$. This follows from the formula 9 on page 525 of this paper by Borel and Hirzebruch:
In principle, the paper gives a recipe for computing $p_k$ for all oriented Grassmannians (and other compact homogeneous spaces), but I did not do the more general calculation.