The formation of self-assembled structures such as micelles has been intensively studied and is well understood. The ability of a solution of amphiphilic molecules to develop micelles is dependent on the concentration and characterized by the critical micelle concentration (cmc), above which micelle formation does occur. Recent studies use a lattice approach in order to determine cmc and show that the correct modeling and analysis of cluster formations is highly nontrivial. We developed a minimalistic coarse grained model for amphiphilic molecules in the continuum and simulated the time evolution via dynamic Monte Carlo simulations in the canonical (NVT) ensemble. Starting from a homogeneous system, we observed and characterized how the initial fluctuations, yielding small aggregates of amphiphilic molecules, end up in the growth of complete micelles. Our model is sufficiently versatile to account for different structures of surfactant systems such as membranes, micelles of variable radius, and tubes at high particle densities by adjusting particle density and potential properties. Particle densities and micellization rates are investigated and an order parameter is introduced so that the dependence of the micellization process on temperature and surfactant density can be studied. The constant density of free particles for concentrations above cmc, e.g., as expected from theoretical considerations, can be reproduced when choosing a careful definition of free volumes. In the cmc regime at low temperatures, different nonequilibrium effects are reported, occurring even for very long time scales.