Protein evolution is a critical component of organismal evolution and a valuable method for the generation of useful molecules in the laboratory. Few studies, however, have experimentally characterized how fundamental parameters influence protein evolution outcomes over long evolutionary trajectories or multiple replicates. In this work, we applied phage-assisted continuous evolution (PACE) as an experimental platform to study evolving protein populations over hundreds of rounds of evolution. We varied evolutionary conditions as T7 RNA polymerase evolved to recognize the T3 promoter DNA sequence and characterized how specific combinations of both mutation rate and selection stringency reproducibly result in different evolutionary outcomes. We observed significant and dramatic increases in the activity of the evolved RNA polymerase variants on the desired target promoter after selection for 96 h, confirming positive selection occurred under all conditions. We used high-throughput sequencing to quantitatively define convergent genetic solutions, including mutational "signatures" and nonsignature mutations that map to specific regions of protein sequence. These findings illuminate key determinants of evolutionary outcomes, inform the design of future protein evolution experiments, and demonstrate the value of PACE as a method for studying protein evolution.