[ginnungagap]

Either you show that your statement P is equivalent to another known independent one or you need to produce a model of P and a model of not P.

The two main techniques to produce new models of ZFC are inner models, that is submodels of a model, for example studying the inner model L (Gödel's constructible universe) is how the consistency of the continuum hypothesis, the generalized continuum hypothesis, the existence of Suslin trees etc. was shown. Looking at L also allows one to conclude that the axiom of choice is consistent with ZF. Another commonly study inner model is the so called HOD, and there's a whole area, "inner model theory" that essentially tries to construct canonical inner models for some statements.

The second big way to produce a model of ZFC is by forcing. This is a very versatile tool that allows to extend a given model of ZFC by adding a new set to it (for example by adding a lot of new reals numbers to a model of CH you can make CH false). Forcing is how the consistency of the negation of CH and GCH was proved.

An interisting example is the negation of AC and its consistency with ZF. If you happen to live in a model of AC every forcing extension will still be a model of AC so it seems that our previous techniques are powerless. But actually what can be done is to look at a carefully chosen submodel of a forcing extension that still models ZF but in which AC fails.