Our solar system, containing the Sun and the planets, is about 2/3 of the way out from the center of the Galaxy. The solar system travels in an orbit around the center of the Galaxy, at a velocity (i.e. speed) of a few hundred kilometers per second, completing one orbit around the center of the Milky Way about every 230 million years. In addition, the solar system is moving at about 20 kilometers per second with respect to the nearby stars. There is also a small amount of motion with respect to the plane of the Galaxy. Currently, the solar system is heading outwards but the gravitational pull of the stars in the galactic plane will eventually cause it to stop and then move back towards the galactic plane.
And yes, our whole Galaxy is also traveling through space. Within the local group of galaxies, the Milky Way's velocity is several hundred kilometers per second.