We present a comprehensive investigation of the dynamics of resonantly excited nonthermal excitons in high-quality GaAs/AlxGa1-xAs multiple-quantum-well structures on picosecond time scales. The dynamics was investigated using the luminescence upconversion technique with two independently tunable, synchronized dye lasers, which allowed measurements of the time evolution of polarized resonant luminescence with 4-ps time resolution. We show that the evolution of excitons from the initial nonthermal distribution to the thermal regime is determined by three different physical processes: (1) the enhanced radiative recombination of the metastable two-dimensional exciton polaritons, (2) the spin relaxation of excitons, and (3) the momentum relaxation of excitons. We also show that these three processes have comparable rates, so that a unified model accounting for all important processes is essential for a correct analysis of the experimental results. Using such a unified model, we have determined the rates of these processes contributing to the initial relaxation of excitons as a function of quantum-well width, temperature, and applied electric field. Quantum confinement strongly influences the radiative recombination and spin relaxation of excitons, and our study provides significant insights into these processes in quantum wells. The measured radiative recombination rate is about a factor of 2 smaller than calculated theoretically. The electric field reduces the electron-hole overlap and hence reduces the spin-relaxation rate of excitons between the optically allowed \+/-1) states. The measured variation is in good qualitative agreement with a recent theory, but somewhat slower than predicted by the theory.