We report an extensive Monte Carlo study of film growth by molecular beam epitaxy. A solid-on-solid model is used on L x L substrate sizes, with periodic boundary conditions. Both deposition and diffusion occur simultaneously; several elementary rules for activated atom hopping are used with only simple interactions between nearest neighbor sites allowed. These rules include preferential or random new-site selection as well as mechanisms which allow or prohibit hopping up step edges. The i nterfaci al width and the kinematic reflection high energy electron diffraction intensity are measured to characterize the roughness of the growing film, and pictures of the surface structure are generated to provide visual information. We find that the hopping rules that initially product small and irregularly shaped two-dimensional clusters result in better layer-by-layer growth whereas rules that result in "smooth" surfaces at early times by producing large compact clusters, become very rough when grown for long times due to islanding. The increase in roughness of the surface over a limited range of deposition time is consistent with that observed in some experiments; however, when atoms are allowed to hog either up or down at step edges, the model suffers from strong crossover behavior and finite-size effects for certain growth conditions, implying the existence of such effects in real systems. At very late times the interfacial width for three of the hopping rules has a logarithmic dependence on time and substrate sizes indicating a behavior similar to the Edwards-Wilkinson model, and from the dynamic scaling analysis of the long-time results we obtain the dynamic exponent of z = 1.61. When all allowed diffusion events are random a e find a power law increase of the interfacial width with time, suggesting that the model is in a different universality class.