We present 5 to 36 micron mid-infrared spectra of 82 young stars in the ~2 Myr old Chamaeleon I star-forming region, obtained with the Spitzer Infrared Spectrograph (IRS). We have classified these objects into various evolutionary classes based on their spectral energy distributions and the spectral features seen in the IRS spectra. We have analyzed the mid-IR spectra of Class II objects in Chamaeleon I in detail, in order to study the vertical and radial structure of the protoplanetary disks surrounding these stars. We find evidence for substantial dust settling in most protoplanetary disks in Chamaeleon I. We have identified several disks with altered radial structures in Chamaeleon I, among them transitional disk candidates which have holes or gaps in their disks. Analysis of the silicate emission features in the IRS spectra of Class II objects in Chamaeleon I shows that the dust grains in these disks have undergone significant processing (grain growth and crystallization). However, disks with radial holes/gaps appear to have relatively unprocessed grains. We further find the crystalline dust content in the inner (< 1-2 AU) and the intermediate (< 10 AU) regions of the protoplanetary disks to be tightly correlated. We also investigate the effects of accretion and stellar multiplicity on the disk structure and dust properties. Finally, we compare the observed properties of protoplanetary disks in Cha I with those in slightly younger Taurus and Ophiuchus regions and discuss the effects of disk evolution in the first 1-2 Myr.