In treating Human Immunodeficiency Virus (HIV) infection, strict adherence to drug therapy is crucial in maintaining a low viral load, but the high dosages required for this often have toxic side effects which make perfect adherence to Antiretroviral Therapy (ART) unsustainable. The imperfect patient adherence to ART and the development of resistant strains in the viral load has led to the development of alternative treatments that incorporate immunological response. This paper investigates theoretically and numerically the effect of immune effectors, such as the cytotoxic lymphocyte (CTL), in modeling HIV pathogenesis; our results suggest the significant impact of the immune response on the control of the virus during primary infection. Qualitative aspects (including positivity, boundedness, stability, uncertainty, and sensitivity analysis) are addressed. Additionally, by introducing drug therapy, we analyze numerically the model to assess the effect of treatment consisting of a combination of several antiretroviral drugs. Nevertheless, even in the presence of drug therapy, ongoing viral replication can lead to the emergence of drug-resistant virus variances. This fact is addressed in our model by including two viral strains, wild-type and drug-resistant. Our results show that the inclusion of the CTL compartment produces a higher rebound for an individual's healthy helper T-cell compartment than does drug therapy alone. Furthermore, we quantitatively characterize successful drugs or drug combination scenarios for both strains of virus.