What determines how fast clouds move?
Cloud particles are suspended in air, and their movement is governed by:


Wind; whichever way the wind blows, the clouds go (with some exceptions such as lenticular clouds; thanks jamesqf comment).  The wind is often strongest in the upper troposphere (there are no clouds in the upper atmosphere), so upper tropospheric clouds will move faster than near-surface clouds.
Convection: strong updrafts will force particles upward, as will forced convection by wind reaching obstacles
Particle size: larger particles are more likely to fall down.  Note that "fall down" may be used either relative to the surrounding air, or relative to the ground (i.e. precipitation that may or may not reach the ground).


So the type of cloud is indirectly relevant: different cloud types are associated with different particle sizes, are to a larger or smaller degree the consequence of convection, and are more or less likely to contain precipitation-sized particles.