I believe the answers to your and related questions are covered in Ch.X of

[1] Foundations of Differential Geometry, vol. II, Kobayashi, Nomizu

More specifically, let $\mathfrak g:=Lie(SU(n))$ and $\mathfrak k:=Lie(K)$ be Lie algebras of the corresponding groups. Since $SU(n)$ is a compact simple Lie group there is unique (up to multiple) invariant metric $\kappa$ on $\mathfrak g$ coming from the Killing form.

Let us fix a point $p=[eK]\in M=SU(n)/K$; any invariant tensor on $M$ is completely determined by its value at $p$. In particular, invariant metric on $M$ is determined by a metric on $T_p M\simeq \mathfrak g/\mathfrak k$, where isomorphism is given by the infinitesimal action along $X\in\mathfrak g$. One of the ways to equip $\mathfrak g/\mathfrak k$ with a metric is to identify it with a $\kappa$-orthogonal complement of $\mathfrak k$. This way you get so-called naturally reductive homogeneous space. For $K=SU(n-1)$ this gives Fubini-Study metric on $\mathbb P(n)$; for $K=S(U(k)\times U(n-k))$ this metric makes $Gr(n,k)$ a symmetric space.

There are two important connections on $M$ with a naturally reductive metric. One is the standard Levi-Civita connection $\nabla^{LC}$ corresponding to the metric defined above, another one is canonical connection $\nabla$ (see Ch X.2 of [1]). Important facts are:

1. $\nabla^{LC}$ and $\nabla$ have the same geodesics (Theorem 2.10 and Theorem 3.3 in [1])
2. Geodesic of $\nabla$ in the direction $X\in \mathfrak k^\perp\simeq T_p M$ is given by the curve $\exp (tX)p$ (Corollary 2.5 in [1]).
3. Any invariant tensor field on $M$ is $\nabla$-parallel.

Finally, I would like to mention that naturally reductive metric is not the only invariant metric on $M$ and other invariant metrics might have different geodesics.

I believe the answers to your and related questions are covered in Ch.X of

[1] Foundations of Differential Geometry, vol. II, Kobayashi, Nomizu

More specifically, let $\mathfrak g:=Lie(SU(n))$ and $\mathfrak k:=Lie(K)$ be Lie algebras of the corresponding groups. Since $SU(n)$ is a compact simple Lie group there is unique (up to multiple) invariant metric $\kappa$ on $\mathfrak g$ coming from the Killing form.

Let us fix a point $p=[eK]\in M=SU(n)/K$; any invariant tensor on $M$ is completely determined by its value at $p$. In particular, invariant metric on $M$ is determined by a metric on $T_p M\simeq \mathfrak g/\mathfrak k$, where isomorphism is given by the infinitesimal action along $X\in\mathfrak g$. One of the ways to equip $\mathfrak g/\mathfrak k$ with a metric is to identify it with a $\kappa$-orthogonal complement of $\mathfrak k$. This way you get so-called naturally reductive homogeneous space. For $K=SU(n-1)$ this gives Fubini-Study metric on $\mathbb P(n-1)$; for $K=S(U(k)\times U(n-k))$ this metric makes $Gr(n,k)$ a symmetric space.

There are two important connections on $M$ with a naturally reductive metric. One is the standard Levi-Civita connection $\nabla^{LC}$ corresponding to the metric defined above, another one is canonical connection $\nabla$ (see Ch X.2 of [1]). Important facts are:

1. $\nabla^{LC}$ and $\nabla$ have the same geodesics (Theorem 2.10 and Theorem 3.3 in [1])
2. Geodesic of $\nabla$ in the direction $X\in \mathfrak k^\perp\simeq T_p M$ is given by the curve $\exp (tX)p$ (Corollary 2.5 in [1]).
3. Any invariant tensor field on $M$ is $\nabla$-parallel.

Finally, I would like to mention that naturally reductive metric is not the only invariant metric on $M$ and other invariant metrics might have different geodesics.