We present the Spitzer/Infrared Spectrograph (IRS) spectra of 157 compact Galactic planetary nebulae (PNe). These young PNe provide insight on the effects of dust in early post-asymptotic giant branch evolution, before much of the dust is altered or destroyed by the hardening stellar radiation field. Most of the selected targets have PN-type IRS spectra, while a few turned out to be misclassified stars. We inspected the group properties of the PN spectra and classified them based on the different dust classes (featureless or F, carbon-rich dust or CRD, oxygen-rich dust or ORD, mixed-chemistry dust or MCD) and subclasses (aromatic and aliphatic, and crystalline and amorphous). All PNe are characterized by dust continuum and more than 80% of the sample shows solid-state features above the continuum, in contrast with the Magellanic Cloud sample where only ~40% of the entire sample displays solid-state features; this is an indication of the strong link between dust properties and metallicity. The Galactic PNe that show solid-state features are almost equally divided among the CRD, ORD, and MCD. We analyzed dust properties together with other PN properties and found that (1) there is an enhancement of MCD PNe toward the Galactic center, in agreement with studies of Galactic bulge PNe; (2) CRD PNe could be seen as defining an evolutionary sequence, contrary to the ORD and MCD PNe, which are scattered in all evolutionary diagrams; (3) carbon-rich and oxygen-rich grains retain different equilibrium temperatures, as expected from models; and (4) ORD PNe are highly asymmetric, i.e., bipolar or bipolar core, and CRD PNe highly symmetric, i.e., round or elliptical; point symmetry is statistically more common in MCD than in other dust class PNe. By comparing the sample of this paper to that of Magellanic Cloud PNe, we find that the latter sample does not include MCD PNe, and the other dust classes are differently populated, with continuity of the fraction of F, CRD, ORD, and MCD populations from high to low metallicity environments. We also find similar sequences for CRD PNe in the Galactic disk and the Magellanic Clouds, except that the Magellanic Cloud PNe seem to attain higher dust temperatures at similar evolutionary stages, in agreement with the observational findings of smaller dust grains (i.e., lower radiation efficiency) in low metallicity interstellar environments. Based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA.