Galaxies move through space with velocities of the order of a several 100 km per second; small velocities for small groups (~100 km/s; e.g Carlberg et al. 2000) and large velocities for rich clusters (~1000 km/s; e.g Girardi et al. 1993).
In addition to this so-called "peculiar velocity", galaxies also also carried away from each other due to the expansion of the Universe, at a velocity proportional to the distance from each other (the "Hubble flow"). But this is not a motion through space; rather it is space itself that is expanding (and hence the velocity may exceed the speed of light for sufficiently large distances).
One may define a "global reference frame" with respect to which velocities are measured. Any reference is valid, but it makes sense to use the frame in which all galaxies are, on average, in rest (when the Hubble flow is subtracted)$^\dagger$. In this frame, the Milky Way has a velocity of 627 ± 22 km/s (Kogut et al. 1993).
$^\dagger$Formally, one uses the frame in which the cosmic microwave background is isotropic.