In this paper, we describe two complementary nonlinear spectroscopy methods which both allow one to achieve Doppler-free spectra of atomic gases. First, saturated absorption spectroscopy is used to investigate the structure of the 5S[subscript 1/2] [right arrow] 5P[subscript 3/2] transition in rubidium. Using a slightly modified experimental setup, Doppler-free two-photon absorption spectroscopy is then performed on the 5S[subscript 1/2] [right arrow] 5D[subscript 5/2] transition in rubidium, leading to accurate measurements of the hyperfine structure of the 5D[subscript 5/2] energy level. In addition, electric dipole selection rules of the two-photon transition are investigated, first by modifying the polarization of the excitation laser, and then by measuring two-photon absorption spectra when a magnetic field is applied close to the rubidium vapour. All experiments are performed with the same grating-feedback laser diode, providing an opportunity to compare different high-resolution spectroscopy methods using a single experimental setup. Such experiments may acquaint students with quantum mechanics selection rules, atomic spectra and Zeeman effect. (Contains 2 footnotes and 8 figures.)