Here's another rather easy way to define a natural Riemannian metric on a Grassmannian $G(k,n)$. Let $V$ be an $n$-dimensional complex vector space with a hermitian inner product. Given any $k$-dimensional subspace $L \subset V$, there is a natural isomorphism $$ T_LG(k,n) = \operatorname{Hom}(L, V/L) = L^* \otimes V/L $$ The hermitian inner product on $V$ induces a natural hermitian inner on $L^*\otimes V/L$. This defines a hermitian metric on $G(k,n)$ that is easily checked to be invariant under the action of $SU(n)$ on $G(k,n)$. What's less obvious from this construction is that the metric is K"ahler.
This approach probably can be extended to flag manifolds, too.
I haven't looked, but I'm pretty sure this is all explained in Griffiths and Harris.