The ultimate goal of any numerical scheme for partial differential equations (PDEs) is to compute an approximation of user-prescribed accuracy at quasi-minimal computational time. To this end, algorithmically, the standard adaptive finite element method (AFEM) and goal-oriented AFEM (GOAFEM) must integrate an inexact solver and nested iterations with discerning stopping criteria balancing the different error components. The algorithms require several fine-tuned parameters in order to make the underlying analysis work. We review recent developments in the field, recall the up-to-date optimal algorithm and investigate the choice of adaptivity parameters for a prototypical GOAFEM example.