Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), a novel virus that belongs to the Coronoviridae family, emerged in December 2019, causing the COVID-19 pandemic in March 2020. Unlike previous SARS and Middle East respiratory syndrome (MERS) outbreaks, this virus has a higher transmissibility rate, albeit a lower case fatality rate, which results in accumulation of a significant number of mutations and a faster evolution rate. Genomic studies on the mutation rate of the virus, as well as the identification of mutations that prevail and their impact on disease severity, are of great importance for pandemic surveillance and vaccine and drug development. Here, we aim to identify mutations on the SARS-CoV-2 viral genome and their effect on the proteins they are located in, in Greek patients infected in the first wave of the pandemic. To this end, we perform SARS-CoV-2 amplicon-based NGS sequencing on nasopharyngeal swab samples from Greek patients and bioinformatic analysis of the results. Although SARS-CoV-2 is considered genetically stable, we discover a variety of mutations on the viral genome. In detail, 18 mutations are detected in total on 10 SARS-CoV-2 isolates. The mutations are located on ORF1ab, S protein, M protein, ORF3a and ORF7a. Sixteen are also detected in patients from other regions around the world, and two are identified for the first time in the present study. Most of them result in amino acid substitutions. These substitutions are analyzed using computational tools, and the results indicate minor or major impact on the proteins’ structural stability, which could probably affect viral transmissibility and pathogenesis. The correlation of these variations with the viral load levels is examined, and their implication for disease severity and the biology of the virus are discussed.