During the Coronavirus disease (COVID-19) pandemic, everyday life has changed; people faced various difficulties and overcame them differently. This study aimed to identify how subjectively named difficulties, the ability to cope with these difficulties, and psychological well-being reported by individuals during the first quarantine of COVID-19 in Lithuania were related to gender, age, and personality traits. Three hundred fifty-one participants aged 18 to 70 years completed an online questionnaire. Results revealed that half of the respondents reported various difficulties, classified into six categories, related to the first COVID-19 quarantine. Furthermore, 51% of the participants indicated poor well-being; the well-being of women and young adults (aged 18–29) was significantly lower compared to the other groups. Those who indicated COVID-19 pandemic difficulties showed higher levels of Neuroticism, and Neuroticism was positively related to the severity of the difficulties. Coping with these difficulties and well-being were positively related to Extraversion, Agreeableness, Conscientiousness, and Openness, and negatively associated with Neuroticism. The manifestation of the particular personality traits in different groups of indicated difficulties is discussed in the article.