As a galaxy evolves, its stars change the amounts (abundances) of elements within it. Thus determining the abundances of these elements in different locations within a galaxy traces its evolution. This dissertation presents abundances of planetary nebulae in our Galaxy and of HII regions in a nearby galaxy (M51). Observations at optical wavelengths dominated such studies in the past. However, abundances determined from infrared lines have the advantages that they are less affected by extinction and the adopted electron temperature. We employ spectra from the Spitzer Space Telescope Infrared Spectrograph and derive abundances for argon, neon, sulfur, and oxygen. These elements are not usually affected by nucleosynthesis in the progenitor stars of planetary nebulae, and thus their abundances trace the amounts of these elements in the progenitor cloud. The abundances of these elements in HII regions trace the amounts of these elements in the interstellar medium today. We do a case study of abundances in the planetary nebula IC 2448, finding that it has subsolar abundances, which indicates that the progenitor star formed out of subsolar material. We also derive abundances and assess the dust properties of eleven planetary nebulae in the Bulge of the Milky Way. We find that the abundances from these planetary nebulae do not follow the abundance trend observed in planetary nebulae in the Disk. This points toward separate evolution for the Bulge and Disk components. Additionally, we find peculiar dust properties in planetary nebulae in the Bulge which indicate that the progenitors of these nebulae evolved in binaries. Finally, we make a pilot study of the abundances in HII regions across the galaxy M51.