We present the results of a Hubble Space Telescope WFPC2 F555W and F814W survey of 69 dwarf elliptical galaxies (dEs) in the Virgo and Fornax Clusters and Leo Group. The V - I colors of the dE globular clusters, nuclei, and underlying field-star populations are used to trace the dE star formation histories. We find that the dE globular cluster candidates are as blue as the metal-poor globular clusters of the Milky Way. The observed correlation of the dE globular cluster systems' V - I color with the luminosity of the host dE is strong evidence that the globular clusters were formed within the halos of dEs and do not have a pregalactic origin. Assuming that the majority of dE clusters are old, the mean globular cluster color-host galaxy luminosity correlation implies a cluster metallicity-galaxy luminosity relation of Z_GC L, which is significantly shallower than the field-star metallicity-host galaxy luminosity relationship observed in Local Group dwarfs (Z_FS L^0.4). The dE stellar envelopes are 0.1-0.2 mag redder in V - I than their globular clusters and nuclei. This color offset implies separate star formation episodes within the dEs for the clusters and field stars, while the very blue colors of two dE nuclei trace a third star formation event in those dEs less than 1 Gyr ago.