Do planets move east to west or west to east across the sky?

Planets move in both directions across the sky. Because the Earth is also moving and is closer than Mars and the outer planets, the direction of motion of those planets sometimes changes as seen from Earth. By convention, their west-to-east motion is called prograde.

As seen from Earth, the outer planets spend the majority of their time in prograde motion. The east-to-west motion is called retrogade and happens as the Earth's orbital revolution catches up with the slower-moving outer orbits and passes them. The planets spend the majority of their motion across the sky in prograde, as seen from Earth.