Compact groups of galaxies provide a unique environment to study the mechanisms by which star formation occurs amid continuous gravitational encounters. We present 2MASS (JHK) andSpitzer IRAC (3.5-8 μm) and MIPS (24 μm) observations of a sample of 12 Hickson Compact Groups (HCGs 2, 7, 16, 19, 22, 31, 42, 48, 59, 61, 62, and 90) that includes a total of 45 galaxies. The infrared colors of the galaxies in this sample span a range of parameter space, and some trends are apparent in the data. The near-infrared colors of the sample galaxies are largely consistent with being dominated by slightly reddened normal stellar populations. There is also some evidence for aK-band excess in a few cases, which likely indicates the presence of hot dust at or near the sublimation temperature associated with active galactic nuclei or star formation activity. Galaxies that have the most significant polycyclic aromatic hydrocarbon and/or hot dust emission (as inferred from excess 8 μm flux) also tend to have larger amounts of extinction and/orK-band excess and stronger 24 μm emission, all of which suggest ongoing star formation activity. We separate the 12 HCGs in our sample into three types based on the ratio of the group H I mass to dynamical mass. We find evidence that galaxies in the most gas-rich groups tend to be the most actively star forming. Galaxies in the most gas-poor groups tend to be tightly clustered around a narrow range in colors consistent with the integrated light from a normal stellar population. We interpret these trends as indicating that galaxies in gas-rich groups experience star formation and/or nuclear actively until their neutral gas is consumed, stripped, or ionized. The galaxies in this sample exhibit a "gap" between gas-rich and gas-poor groups in infrared color space that is sparsely populated and not seen in theSpitzer First Look Survey sample. This gap may suggest a rapid evolution of galaxy properties in response to dynamical effects. These results suggest that the global properties of the groups and the local properties of the galaxies are connected.