Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) was first detected in humans more than two years ago and caused an unprecedented socio-economic burden on all countries around the world. Since then, numerous studies have attempted to identify various mechanisms involved in the alterations of innate and adaptive immunity in COVID-19 patients, with the ultimate goal of finding ways to correct pathological changes and improve disease outcomes. State-of-the-art research methods made it possible to establish precise molecular mechanisms which the new virus uses to trigger multisystem inflammatory syndrome and evade host antiviral immune responses. In this review, we present a comprehensive analysis of published data that provide insight into pathological changes in T and B cell subsets and their phenotypes, accompanying the acute phase of the SARS-CoV-2 infection. This knowledge might help reveal new biomarkers that can be utilized to recognize case severity early as well as to provide additional objective information on the effective formation of SARS-CoV-2-specific immunity and predict long-term complications of COVID-19, including a large variety of symptoms termed the ‘post-COVID-19 syndrome’.