Drivers and cyclist can get to Mljet by taking ferry from Prapratno on Peljesac peninsula (South of Ploce and Makarska, you'll see the signs) which has frequent connections with mainland.
Updates to price and schedule is on Jadrolinija web site. It takes almost 40 minutes ride by ferry before you're on Mljet and you'll disembark in port 2 kilometers from Sobra. Buy ticket at the office before you'll embark the car on ferry, ticket control is by the staff at pier in front of ferry.
Others can take catamaran from Dubrovnik (Nona Ana) and get on Mljet either on Sobra or in Polace. For Sobra if you want to see a little of interior of island, for Polace if you focus entire stay on National park area.
Local bus connects other towns on Mljet when ferry and catamaran arrive on Mljet but other than that it's not really favourable option to get around, so you'd better rent something (in Polace) to get around.
There is ferry line from Prapratno (Peljesac) to Sobra (Mljet island) every day, 4-5 times a day.
The price for one person, one way is 30 kn (~4 €) bikes are 35 kn (~5 €), motorcycles 60-100 kn (~8-14 €) and if you want to get the car, it will cost you 140 kn (~20 €).
These were the prices while we were visiting. I don't think they changed much, but I attached website of the ferry company so check for yourselves, just in case.