There is a car ferry from Prapratno close to Ston going to Sobra on Mljet. The crossing time is 45 minutes.
If you want to get directly to the national park and leave your car on Peljesac, there is a catamaran from Trstenik half way up to the end of the peninsular. It only runs during high season and it will take you directly to the main entrance of the park in Polace. There is also a catamaran operating from Dubrovnik.
Mljet is the greenest island of the Croatian Adriatic. 72% of the island is covered with forest. Primarily pine woods.
The one place you must visit on Mljet is the lakes at the western part. They are surrounded by pine forests and a lovely environment where many people spend more than a day just walking around.
Go to Polace and buy an entrance ticket at the kiosk in front of the Roman Palace. In Polace there are several restaurants and bars where you can choose to rent bicycles or use the minibus to take you into the park. Entrance fee includes use of minibus and ferry to the Benedictine Monastery on the lake. The island is lovely spot where you can spend some time and maybe have a meal or a drink in the restaurant.
Leaving the island you should take the boat towards Mali Most. Have a walk back from here (1.5 km) or stay on the boat that will then take you back to where you started and where you may catch the minibus again back to Polace.
Cars are not allowed inside the park, but you may drive up to a parking place and walk the last 500 meters to the same spot where the minibus would drop you off.
There is a trail down from Babino Polje marked with red dots and arrows - but be warned! It is tough. Bring water, cover your head from the sun, wear good shoes with rubber soles and expect to use at least two hours. The locals will tell you that you can reach the cave in 30 minutes and that is just possible if you are young and healthy, but you also have to come back and I can tell you from experience that it is very tough to climb the rocks again. I was reminded of my age and lack of regular exercise.
If you want to explore the other end of Mljet you should visit the south-eastern tip of the island and a place called Saplunara. It is known for its sandy beaches. There are two beaches, Big Saplunara and Small Saplunara. Another possibility is to visit a beautiful sandy beach called Blace Bay about 20 minutes walking from Saplunara. It faces the open sea and is shielded by cliffs. The beach is about 1 km long and surrounded by old pines. Blace may be used as a naturist beach.
When you go back again from Sobra be aware that the cars will get on board in the order they arrive. Just like all other ferry lines between the islands it is not possible to book. You just drive up, buy your ticket at the office and wait. In June 2006 there were very few cars, but I have no idea how popular it will be during July and August. It could be a good idea to ask at the office how early they advice you to queue up.
Part of Mljet was declared a National Park in 1960.
You can check the coordinates (GPS and maps) for these places below:
Coordinates