Depending on the resort you choose and if you are lucky, you get to Tenggol island by speedboat from Kuala Dungun jetty but Tenggol Island Resort arranged a wooden bum/dive boat for us.
The trip from Kuala Dungun to the island will take about 1 hour by speed boat and 2 hours for bum boat.
You can take a bus to Dungun from the Putra or Pudu bus stations in KL. I reckon it costs less that RM30 nowadays. For punctuality and safety, I recommend Transnasional though it doesn't offer the most luxurious coaches. The bus will pass by Dungun, so you'll have to tell the driver where you want to get down. If you have never been to Dungun before, tell the driver beforehand so he'll inform you when you reach. Don't forget to get off the bus coz the bus journey continues to Kuala Terengganu, 90 km north of Dungun.