Lots of bacteria are mobile.
Ever hear of an ameba?
Or a flagellum?
Amoeba, strictly speaking, are not “bacteria”, but a somewhat more sophisticated organism. They are able to move by extending a pseudopod, then flow their internal vital protoplasm into that pseudopod. Similarly, they “eat” their sustenance, usually some form of bacteria, or dead organic material, by encapsuling the prey or “dinner” and absorbing it into their protoplasm.
Flagellum is a structure, not any kind of genus, and it may be found on certain species of bacteria, which have developed a rudimentary means of propulsion by the actions of the whip-like “tail” that seems to be capable of a kind of volitional movement. This structure is also found on other organisms, unrelated to bacteria, sort of an analog of the tail of something like a tadpole.