For me, checking out locally produced tea is a must, wherever I am in China. It's nice to mentally slow down and sit for an hour at the tea merchant's hand-carved tea table, sampling a variety of tea.
Hainan Specialty Tea Market is a good spot to do just that. You can sample teas from all over China, but be sure to try some native to Hainan. The three most well-known are Shuiman tea (水满茶), Xianglan tea (香兰茶), and coconut tea.