Controlling the dynamics of spins on surfaces is pivotal to the design of spintronic(1) and quantum computing(2) devices. Proposed schemes involve the interaction of spins with graphene to enable surface-state spintronics(3,4) and electrical spin manipulation(4-11). However, the influence of the graphene environment on the spin systems has yet to be unravelled(12). Here we explore the spin-graphene interaction by studying the classical and quantum dynamics of molecular magnets(13) on graphene. Whereas the static spin response remains unaltered, the quantum spin dynamics and associated selection rules are profoundly modulated. The couplings to graphene phonons, to other spins, and to Dirac fermions are quantified using a newly developed model. Coupling to Dirac electrons introduces a dominant quantum relaxation channel that, by driving the spins over Villain's threshold, gives rise to fully coherent, resonant spin tunnelling. Our findings provide fundamental insight into the interaction between spins and graphene, establishing the basis for electrical spin manipulation in graphene nanodevices.