Take the bus bus from Surat Thani to Krabi . There are minivans, as well. I'm not sure where the bus station is (can't remember), but Krabi is small and just about everything is within a short walking distance.
Then there are two options for getting to Koh Jum from Krabi (that I know of.) You could take the ferry to Koh Lanta and tell them you want to go to Koh Jum. A longtail boat will meet the ferry off the coast of Koh Jum and will take you back to the island where the boat will run up onto the beach. The ferry leaves Krabi from the Saphan Jao Fah pier right in town.
The other way is to go to Ban Laem Kruat and get a boat from there. From this village you can go to either Koh Jum or Koh Si Boya. It's a bit of a hike there, but if you go to one of the dozens of shops selling boat/train/bus/minivan tickets, arrangements can be made to get you there. Just tell the person selling tickets that you want to go to Koh Jum and they'll make all the arrangements.
If you leave really early in the morning from Surat Thani you might be able to get the afternoon boat that leaves at around 1:30. If you miss it, option #2 will get you there in the afternoon.
Surat Thani to the island should take about 7 or 8 hours. Someone else might have a better idea, though.
I've done both and preferred the longer boat trip, though the village was interesting.