My question is if you have a vector field whose integral curves are geodesic, does it imply that vector field is also killing? It seems like it is, just wanted to make sure if it was indeed true.
In particular if I have a smooth function $f$ such that $|\nabla f|=1$, I know that its integral curves are geodesics. I want to verify that $\nabla f$ is also killing.
No. Consider $f(x) = |x|$ in Euclidean $n$-space, $n \ge 2$: its gradient is the radial unit vector field, whose integral curves are the radial rays. As you flow along this field, areas will be stretched in the angular direction, so it certainly does not generate isometries.