We present a comprehensive study on the impact of the environment of compact galaxy groups on the evolution of their members using a multi-wavelength analysis, from the UV to the infrared, for a sample of 32 Hickson compact groups (HCGs) containing 135 galaxies. Fitting the SEDs of all galaxies with the state-of-the-art model of da Cunha (2008) we can accurately calculate their mass, SFR, and extinction, as well as estimate their infrared luminosity and dust content. We compare our findings with samples of field galaxies, early-stage interacting pairs, and cluster galaxies with similar data. We find that classifying the groups as dynamically old or young, depending on whether or not at least one quarter of their members are early-type systems, is physical and consistent with past classifications of HCGs based on their atomic gas content. [...ABRIDGED...] We also examine their SF properties, UV-optical and mid-IR colors, and we conclude that all the evidence point to an evolutionary scenario in which the effects of the group environment and the properties of the galaxy members are not instantaneous. Early on, the influence of close companions to group galaxies is similar to the one of galaxy pairs in the field. However, as the time progresses, the effects of tidal torques and minor merging, shape the morphology and star formation history of the group galaxies, leading to an increase of the fraction of early-type members and a rapid built up of the stellar mass in the remaining late-type galaxies.
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