Q: Can black holes move through space?
A: Black holes can indeed move through space. The really massive black holes at the centers of galaxies will stay there unless something catastrophic happens, like a direct collision between two galaxies.
The much smaller black holes formed from the explosions of stars can move fairly quickly because they receive a kick from the explosion. See, for example:
Astronomers know about the rapid motion of this black hole because it has a nearby companion star that's slowly being pulled apart by the gravity of the black hole. Rapidly moving black holes that are flying solo will be essentially impossible to detect, since space is very big and they will encounter other objects only very rarely. Only the most massive stars produce black holes, and such stars are rare, so there's no need to worry about a rogue black hole heading towards the Solar System.