Is there any direct bus or train to Sanya or Haikou from Nanning?

Yes, you can take a direct bus to Haikou from Jiangnan Passenger Transport Station. It's located at No. 236, Xingguang Street, Jiangnan District.
The buses are scheduled at 9:50, 14:30, 15:00, 16:00, 17:00, 18:40, and 19:40. The trip takes about 10 hours. And the fare is CNY 230 per person.
After arriving, you can take a bullet train to Sanya (about 2 hours).