Using atomic force microscopy (AFM), supported by semicontinuum numerical simulations, we determine the effect of tip-subsurface van der Waals interactions on nanoscale friction and adhesion for suspended and silicon dioxide supported graphene of varying thickness. While pull-off force measurements reveal no layer number dependence for supported graphene, suspended graphene exhibits an increase in pull-off force with thickness. Further, at low applied loads, friction increases with increasing number of layers for suspended graphene, in contrast to reported trends for supported graphene. We attribute these results to a competition between local forces that determine the deformation of the surface layer, the profile of the membrane as a whole, and van der Waals forces between the AFM tip and subsurface layers. We find that friction on supported monolayer graphene can be fit using generalized continuum mechanics models, from which we extract the work of adhesion and interfacial shear strength. In addition, we show that tip-sample adhesive forces depend on interactions with subsurface material and increase in the presence of a supporting substrate or additional graphene layers.