Amyloid fibrils are insoluble mainly β-sheet aggregates of proteins or peptides. The multi-step process of amyloid aggregation is one of the major research topics in structural biology and biophysics because of its relevance in protein misfolding diseases like Alzheimer's, Parkinson's, Creutzfeld-Jacob's, and type II diabetes. Yet, the detailed mechanism of oligomer formation and the influence of protein stability on the aggregation kinetics are still matters of debate. Here a coarse-grained model of an amphipathic polypeptide, characterized by a free energy profile with distinct amyloid-competent (i.e. β-prone) and amyloid-protected states, is used to investigate the kinetics of aggregation and the pathways of fibril formation. The simulation results suggest that by simply increasing the relative stability of the β-prone state of the polypeptide, disordered aggregation changes into fibrillogenesis with the presence of oligomeric on-pathway intermediates, and finally without intermediates in the case of a very stable β-prone state. The minimal-size aggregate able to form a fibril is generated by collisions of oligomers or monomers for polypeptides with unstable or stable β-prone state, respectively. The simulation results provide a basis for understanding the wide range of amyloid-aggregation mechanisms observed in peptides and proteins. Moreover, they allow us to interpret at a molecular level the much faster kinetics of assembly of a recently discovered functional amyloid with respect to the very slow pathological aggregation.