In the Local Group, almost all satellite dwarf galaxies that are within the virial radius of the Milky Way (MW) and M31 exhibit strong environmental influence. The orbital histories of these satellites provide the key to understanding the role of the MW/M31 halo, lower-mass groups, and cosmic reionization on the evolution of dwarf galaxies. We examine the virial-infall histories of satellites with \({M_{\rm star}}=10^{3-9} {~\mbox{M}_\odot}\) using the ELVIS suite of cosmological zoom-in dissipationless simulations of 48 MW/M31-like halos. Satellites at \(z=0\) fell into the MW/M31 halos typically \(5-8 {~\mbox{Gyr}}\) ago at \(z=0.5-1\). However, they first fell into any host halo typically \(7-10 {~\mbox{Gyr}}\) ago at \(z=0.7-1.5\). This difference arises because many satellites experienced “group preprocessing” in another host halo, typically of \({M_{\rm vir}}\sim 10^{10-12} {~\mbox{M}_\odot}\), before falling into the MW/M31 halos. Satellites with lower-mass and/or those closer to the MW/M31 fell in earlier and are more likely to have experienced group preprocessing; half of all satellites with \({M_{\rm star}}< 10^6 {~\mbox{M}_\odot}\) were preprocessed in a group. Infalling groups also drive most satellite-satellite mergers within the MW/M31 halos. Finally, none of the surviving satellites at \(z=0\) were within the virial radius of their MW/M31 halo during reionization (\(z > 6\)), and only \(< 4\%\) were satellites of any other host halo during reionization. Thus, effects of cosmic reionization versus host-halo environment on the formation histories of surviving dwarf galaxies in the Local Group occurred at distinct epochs and are separable in time.