The Milky Way is believed to be moving in the direction of the Great Attractor, which is a gravity anomaly in intergalactic space within the range of the Centaurus Supercluster.
Astronomers also believe the Milky Way is moving at about 630 km per second, although we must keep in mind the absolute velocity of any object through space is relative.
The Milky Way must rotate in order to create its spiral shape, otherwise gravity would make all the particles merge into more of a spherical shape.
While the galaxy itself rotates on its axis, sort of rushing head over heels through space, all the stars within the galaxy rotate around its center.
Our solar system takes one galactic year, about 225 million years, to make one revolution around the center of the galaxy.