Recently, the tribological properties of graphene have been intensively examined for potential applications in micro- and nano-mechanical graphene-based devices. Here, we report that the tribological properties can be easily altered via simple chemical modifications of the graphene surface. Friction force microscopy measurements show that hydrogenated, fluorinated, and oxidized graphenes exhibit, 2-, 6-, and 7-fold enhanced nanoscale friction on their surfaces, respectively, compared to pristine graphene. The measured nanoscale friction should be associated with the adhesive and elastic properties of the chemically modified graphenes. Density-functional theory calculations suggest that, while the adhesive properties of chemically modified graphenes are marginally reduced down to similar to 30 %, the out-of-plane elastic properties are drastically increased up to 800 %. Based on these findings, we propose that nanoscale friction on graphene surfaces is characteristically different from that on conventional solid surfaces; stiffer graphene exhibits higher friction, whereas a stiffer three-dimensional solid generally exhibits lower friction. The unusual friction mechanics of graphene is attributed to the intrinsic mechanical anisotropy of graphene, which is inherently stiff in plane, but remarkably flexible out of plane. The out-of-plane flexibility can be modulated up to an order of magnitude by chemical treatment of the graphene surface. The correlation between the measured nanoscale friction and the calculated out-of-plane flexibility suggests that the frictional energy in graphene is mainly dissipated through the out-of-plane vibrations, or the flexural phonons of graphene.