I have two tricks: Dehn filling and drilling. I've used the former to study subgroup separability, as a technical trick to reduce the proof of tameness of Kleinian groups in the cusped case to the non-cusped case, to produce non-Haken 3-manifolds, as well as study exceptional (non-hyperbolic) Dehn fillings on a cusped manifold. I've used drilling also in the proof of tameness, to relate the volume of closed hyperbolic manifolds to cusped ones, and in the solution of Simon's conjecture about epimorphisms between knot groups.

As you might guess, these are really the same trick (one is the inverse operation of the other), but I like to think of them as two ;).