Which travels planet or the sun?

Answer 1

In our solar system, planets travel faster because their mass is much smaller than the Sun, so they orbit the Sun while the latter remains relatively in the same place. It is more accurate to say both the planets and the Sun orbit a common center of gravity - but in this context, the Sun moves less.

It is of note that the solar system itself is also in a long slow orbit around the galactic center - so in a real sense, the entire solar system is moving; the Milky Way galaxy also shares an orbit around a galactic cluster, and on a larger scale, also within a supercluster, all of which have their own motion.