A phase-stabilized femtosecond laser comb is directly used for high-resolution spectroscopy and absolute optical frequency measurements of one- and two-photon transitions in laser-cooled 87Rb atoms. Absolute atomic transition frequencies, such as the 5S1/2 F=2-->7S1/2 F"=2 two-photon resonance measured at 788,794 768,921 (44) kHz, are determined without a priori knowledge about their values. Detailed dynamics of population transfer driven by a sequence of pulses are uncovered and taken into account for the measurement of the 5P states via resonantly enhanced two-photon transitions.