A lot of plumbers can probably replace a bath vanity, although not many of them would want to do a kitchen cabinet. A good handyman can probably do both. Not to discredit my fellow plumbers on the site. Also, most plumbers probably know a contractor who can do the cabinet. The best thing to do is to ask around and get references.

Other damage could potentially be mold in the wall, damage to sub-floor if it is wood, and damage to wood flooring.

What area are you in, there may be somebody who can point you in the right direction.
Al

First: fix the leak (plumber)
Second: open up walls, floors, check for mold. If mold present, call mold removal expert and have him to sanitize it for you
Third: hire finish carpenter to repair cabinets and trim. You may get lucky; some finish carpenters will repair sub-floor and hard-wood floors. Otherwise, hire floor installer.
Fourth: hire painter to do paint touch ups.