Most practicing mathematicians, when dealing with all the number systems you mention, probably don't really care about the foundations and consistency issue.
To answer your questions: One can start with a model for set theory and then build inside that model a model for what is called Peano Arithmetic (PA). Basically, PA is a set of axioms about the basics of the system of natural numbers. Taken together with the principle of induction (which is not a first order notion) it can easily be shown that PA + Induction is categorical: any two models satisfying these axioms are isomorphic (meaning they are essentially the same).
With a model for PA+Induction one can construct, still within the model of set theory, a model of the integers, and then the rational, by considering an appropriate equivalence relation on sets of pairs of naturals (resp. integers). With a model for the rational now at hand, one can proceed (in several different ways) to complete it and obtain a model of the real numbers. Finally, considering pairs of real numbers, one can define a model for the complex numbers.
You will notice that all of these constructions have the common theme: From an existing model of some system X, construct (somehow) a model for system Y. This shows what is known as relative consistency: If system X is consisten (i.e., has a model) then system Y is consistent as well.
This works quite well but of course relies on the existence of a model for set theory. This is pretty much the end of the road though as it is a fundamental result in logic that a system rich enough to discuss set theory, if consistent can't prove its own consistency. So basically, we all believe that there exists a model for set theory.
There are some variations on the theme. For instance, instead of set theory as foundations one can use Topos theory. In a rich enough topos one can give analogous definitions of the natural, integer, rational, real, and complex numbers and, depending on the topos, these systems can be very different than the classical ones. For instance, there exists a topos in which every real valued functions $f:\mathbb R \to \mathbb R$ is continuous.