Recently, Bi₂Se₃ was added to a new class of materials known as topological insulators. While several studies have provided tantalizing hints towards novel physical properties, such as backscatter suppression and spin-polarized transport, several concerns remain in actual materials. In particular, high defect densities, strong surface band bending, and potential fluctuations have been observed. Here, scanning tunneling microscopy and spectroscopy are used to reveal surface effects in Bi₂Se₃ and CuxBi₂Se₃. First, a detailed examination of defects in bulk-grown samples is described. Then, I provide an analysis of molecular beam epitaxy results, done in collaboration with colleague Yuxuan Chen. Following this, I provide a detailed study of individual point defects in Cu-doped Bi₂Se₃ and examine how Cu is incorporated into the Bi₂Se₃ lattice. Finally, through spectroscopic analysis, a novel depth-sensitive measurement of the local band bending field is developed. Furthermore, for the first time, fluctuations of the Dirac point can be correlated to specific near-surface defects, namely Se vacancies. These analyses provide valuable insights into the preparation of future samples for the investigation of topological insulators. / text