As
siren already pointed out, there will be a
compression
wave (
longitudinal wave) travelling down the rod when you move it.
The velocity at which this happens (and thus the limiting speed for
communication) will be the
speed of sound in the material the rod
is made from. This will definitely be a
lot slower than the
speed of light; for instance the
speed of sound in
iron and
aluminum are approximately 5100 m/s.
For example, hit a nail with a hammer; it will still take a
fraction of a second before the tip "knows" that it should move.