Monolayer transition metal dichalcogenides (TMDs) are characterized by strong light-matter interactions due to enhanced excitonic effects, which make them exciting materials for fundamental physics and optoelectronics applications. Moreover, the valley-dependent chirality of the band structure in TMDs significantly modifies the optical selection rules for single- and multiphoton processes. Here, we propose an analytical approach for calculating the linear and nonlinear optical (NLO) responses of monolayer TMDs, including excitonic effects at low photon energies. Based on this approach, we provide an informative diagram which encompasses all excitonic selection rules. The diagram enables us to identify main transitions for the first-, second-, and third-order optical responses. As a case study, we calculate the optical conductivity and second-/third-harmonic generation responses of monolayer MoS2 and demonstrate that the analytical approach accurately reproduces the spectra obtained using the Bethe-Salpeter equation (BSE). Moreover, the analytical approach enables us to obtain valuable physical insight into the fundamental transitions responsible for individual resonances, which is not straightforward in the full BSE method. Our analytical approach can readily be extended to higher-order nonlinearities and, hence, provides a simple but accurate tool for analyzing experimental NLO spectra of monolayer TMDs.