Mljet offers a unique setting: an isolated island, less tourists than anywhere else, and amazing nature. The island has two ports where the ferries come in: Polace and Sobra. I highly recommend staying in Polace, the gateway to Mljet National Park. This is where I was based. You can easily hire bicycles or scooters and follow the trails through the incredible nature that surrounds you.
In the heart of the national park, there's an inland lake. The island offers a fine choice of walks, a monastery, and a high hill overlooking the landscape which reveals an incredible view of the island and its coasts. While I was absolutely delighted with my stay in Mljet, I wasn't overly impressed by the lack of an old town. Polace is like a new, unexciting city.