Background: Coronavirus disease (COVID-19) has affected people’s lives globally. While important research has been conducted, much remains to be known. In Bangladesh, initial treatment (self-administered, hospitalized), persistent COVID-19 symptoms (“long COVID-19”), and whether COVID-19 leads to changes in mental state, such as depressive symptoms, of people are not known. This study aimed to examine treatment, persistent symptoms, and depression in people who had been infected with COVID-19 in Bangladesh. Methods: A cross-sectional survey was conducted on 1002 individuals infected with COVID-19 (60% male; mean age = 34.7 ± 13.9; age range = 18–81 years), with data taken over a one-month period (11 September 2020 to 13 October 2020). A self-reported online questionnaire was used to collect data on socio-demographics, lifestyle, COVID-19 symptoms (during and beyond COVID-19), medication (over-the-counter or doctor-prescribed), and depression (assessed using the Patient Health Questionnaire (PHQ-9)). Results: Twenty-four percent of participants self-medicated with over-the-counter medicine when they were first diagnosed with COVID-19. Self-medication was higher among female vs. male respondents (29.6% vs. 20.2%, respectively, p = 0.002). A minority (20%) reported that they experienced persistent COVID-like symptoms after recovering from COVID-19. The most reported persistent symptoms were diarrhea (12.7%) and fatigue (11.5%). Forty-eight percent of participants were categorized as having moderate to severe depression. Based on multivariate regression analysis, depression during COVID-19 was positively associated with lower family income, poor health status, sleep disturbance, lack of physical activity, hypertension, asthma/respiratory problems, fear of COVID-19 re-infection, and persistent COVID-19 symptoms. Conclusions: The findings suggest a need for appropriate interventions for COVID-19 patients to promote physical and mental wellbeing.