The main ferry connection to Mljet runs from Dubrovnik to Sobra, Mljet's principal port on the eastern end of the island. Jadrolinija operates this route year-round, with more frequent sailings in summer. From Sobra it's about 20km west to Soline by road — you'll need a car, a bike, or patience with infrequent local buses. The journey through the island's pine forests takes about 30 minutes by car.
A faster option in summer is the Nona Ana catamaran from Dubrovnik, which docks at Polače on the western end of the island — just a few kilometres from Soline. From Polače, the road to Soline passes through the national park and is flat enough to cycle. Alternatively, a car ferry runs from Prapatno on the Pelješac peninsula to Sobra, which is useful if you're arriving from the north by road. Visitors entering the national park pay an entrance fee at the gate; this covers use of the park's facilities including the island boat service to the monastery.