A nice geometric way of endowing a Grassmann manifold with a metric (understood here as a distance, and not directly as a Riemannian metric) is to use the Hausdorff distance for subsets of the round sphere.
Consider $V$ a real vector space of dimension $n$ endowed with an inner product, and let $Gr_k(V)$ be the Grassmannian of $k$-planes on $V$. Let $x,y\in Gr_k(V)$, and denote by $S_x$ and $S_y$ the subspheres of the unit sphere $S_V=\{v\in V:\|v\|=1\}$ defined by intersecting it with the subspaces $x$ and $y$ respectively. Then $S_x,S_y\subset S_V$ are closed subsets and the distance between $x,y\in Gr_k(V)$ can be defined as the Hausdorff distance between these sets: $$dist(x,y)=dist_H(S_x,S_y).$$ Although I have not checked the details, this probably agrees with the distance induced by the homogeneous Riemannian metric discussed in the other posts and comments; but to me it seems easier to visualize.