Graphene, a two-dimensional nanomaterial with unique biomedical properties, has attracted great attention due to its potential applications in graphene-based drug delivery systems (DDS). In this work graphene sheets with various sizes and graphene oxide functionalized with polyethylene glycol (GO-PEG) are utilized as nanocarriers to load anticancer drug molecules including CE6, DOX, MTX, and SN38. We carried out molecular dynamics calculations to explore the energetic stabilities and diffusion behaviors of the complex systems with focuses on the effects of the sizes and functionalization of graphene sheets as well as the number and types of drug molecules. Our study shows that the binding of graphene-drug complex is favorable when the drug molecules and finite graphene sheets become comparable in sizes. The boundaries of finite sized graphene sheets restrict the movement of drug molecules. The double-side loading often slows down the diffusion of drug molecules compared with the single-side loading. The drug molecules bind more strongly with GO-PEG than with pristine graphene sheets, demonstrating the advantages of functionalization in improving the stability and biocompatibility of graphene-based DDS.