If one wants to show "If A, then B", first find an interesting consequence C of B that looks easier to prove than B itself, but not so simple that it can be immediately deduced from A. Then try to prove "If A, then C". Finally, show "If A and C, then B". The point is that this factors the original problem into two simpler ones. If one believes that the original implication is true, then the two sub-implications must be true also, so one is not "losing" anything by trying this method.
A variant approach, once one has successfully obtained "If A, then C", is to deconstruct that proof in order to find out what made it work; this can then give valuable clues as to how to then prove the stronger statement "If A, then B". This tends to work well when C acts as a "simplified toy model" of B.
For instance, if trying to prove an identity of the form , where is some expression depending on a parameter , and is independent of , one might first try to establish the simpler statement " is independent of ", and then to establish for some special value of (note that could be an asymptotic value, such as infinity).
Note that many proofs of identities by induction are basically of this form.
This post is based on this Google Buzz article by Terence Tao.