In the era right after the civil war, Americans used to spend their leisure time by reading books, specially novels, and newspapers which is still a common activity in today's time.
Other than that, they took a keen interest in arts and history and visited museums to explore more about it. Amusements parks were also an attraction for them.
Shows like Wild West (travelling theoretical performance show) were very popular back then which are equivalent to today's concert and theater shows. Most of these activities are very similar that people of this time like to practice in their leisure time.