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.
1.Set-theoretic technique : forcing, e.g. Cohen forcing which was used in the proof of the independence of the continuum hypothesis.A good reference is Shelah's book on forcing.
2. Model-theoretic: construct a model for ZFC in which the negation of the statement is true and another model for which the statement is true.
There may be proof-theoretic techniques based on the non-existence of proofs involving finitely many steps, but I am not aware of these.
Since ZFC can't prove its own consistency, you can't actually prove any statement is independent (an inconsistent system proves everything). Instead what is proved is that if ZFC is consistent then the statment is independent. There's a theorem called Gödel's completeness theorem (a somewhat confusing name in the light of his more famous incompleteness theorem, but they talk about different things so there's no conflict between them) that says that a system is consistent if and only if there exists a model of that system. So the way that you normally prove "If ZFC is consistent then S is independent" is by assuming ZFC is consistent, using the Completeness Theorem to show that it has a model, altering that model to produce models of ZFC+S and ZFC+¬S, and then using the Completeness Theorem in the other direction to conclude that both ZFC+S and ZFC+¬S are consistent.
To show that a sentence P is independent of a theory (that is, a set of sentences) T, generally one constructs or demonstrates the existence of a model of T + P and also produces a model of T + (not P).
Show that it implies the consistency of ZFC, and that its negation also does so. Or show that it's equivalent to a statement that's already on that list.
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.