The dissipative standard map is studied in both the regular and chaotic regimes using a numerical technique that allows us to obtain the stable and unstable periodic orbits of this map in a systematic way. The method applies in all limits of the map, including the Hamiltonian limit (where the dissipation parameter is b=1) and the strongly dissipative limit (b=0), where the map becomes the one-dimensional circle map. Using our technique we calculate the phase diagram of the map both below the critical line, where the dynamics is dominated by mode locking to stable periodic orbits, and above it, where chaotic behavior appears. We then study the scaling properties of the Arnold tongues as the coefficient of the nonlinear term, V, increases from zero. We find a scaling relation between the width of the Arnold tongue (with w=p/q) and the Liapunov exponent of the most stable orbit in that tongue, which both scale like Vq. This result is obtained numerically and confirmed by an analytic argument.