I work and live in China mainland with my family. I have Z visa / residence permit.

Yes, you are perfectly allowed to travel to Hainan in the same way as you would be traveling to any other part of mainland China. You are allowed to travel to Hainan by every available means of transport including the train, ferry or plane. This is, in your case, just another mainland Chinese domestic travel and you don't need any other documents or permits with the exception of your travel tickets :)