Effective management of the global pandemic caused by Severe Acute Respiratory Syndrome Coronavirus 2 (also known as COVID-19), resulted in the implementation of severe restrictions in movement and enforcement of social distancing measures. This study aimed to understand and characterize the psychosocial effects of the COVID-19 pandemic in the general population and to identify risks and protective factors that predict changes in mental health status. In addition, the study investigated compliance with precautionary measures (PM) to halt the spread of the virus. The online anonymous survey collected information on sociodemographic data, compliance with PM, quality of life (QOL), and mental health via the Generalized Anxiety Disorder-7 (GAD-7) and Patient Health Questionnaire-9 (PHQ-9). A total of 1642 adult participants (71.6% women, 28.4% men) completed the survey in the European island country, Cyprus. A large percentage (48%) reported significant financial concerns and 66.7% significant changes in their QOL. About 41% reported symptoms associated with mild anxiety; 23.1% reported moderate-severe anxiety symptoms. Concerning depression, 48% reported mild and 9.2% moderate-severe depression symptoms. Women, younger age (18–29), student status, unemployment status, prior psychiatric history, and those reporting greater negative impact on their QOL, were at higher risk for increased anxiety and depression symptoms (p < 0.05). The youngest age group and males also reported lower levels of compliance with PM. Higher compliance with PM predicted lower depression scores (p < 0.001) but higher anxiety for measures related to personal hygiene. The results of this study provide important data on the effects of the COVID-19 outbreak on mental health and QOL and identify a variety of personal and social determinants that serve as risks and protective factors. Furthermore, it has implications for policy makers demonstrating the need for effective mental health programs and guidance for the implementation of PM as a public health strategy.