Another next thing you will absolutely enjoy doing is cycling - and since you're not allowed to enter lake areas by car or motorbike unless you're guest somewhere within its vicinity, it is great way to cover the distances. If you like doing it faster than walking, of course. Those paved roads only allow to take mini buses from NP office (beside guests who stay in Soline for example) so it's really and mostly quiet, safe and easy.
If you're already thinking about renting a bike you'll get them lot cheaper in Polace by the ticket office and its only few kilometers to the lake area anyways, and slopes aren't too dificult to bike on. Once when you're already by the lakes (where the parking ground is) you'll pay expensive 40 KN per hour. If you had the bike in Polace you'd pay 50 KM for a whole day (in sept 2010). It's worth to consider!
Equipment: You can rent a bike (or bring your own) in Polace, Pomena or by the lakes. These are small distances to cover, not too steep here. Take helmet if you feel safer and water - and don't forget to dispose trash in the right trash bin or area, you don't want to see plastic everywhere (and decaying toilet papers in bushes).
Yes - Mljet National park is to do quiet water sports: swimming, snorkeling, kayaking and canoeing. You will be able to rent boats by the entry to NP by parking place or your guesthouse staff will be able to help you with it.
In the lakes area you're only alowed to do these sports, no motorized ones (which is a great thing). You are not allowed to dive on your own - for this you must get in contact with the NP staff and they will tell you more about it. Also you're not allowed to do fishing in NP area (at least you need permision from the NP office).
Lakes have many excellent natural areas for swimming, shaded with pines and different type of rock and beaches, shallow in the shore or deep straight at it, some for naturist, other family oriented - whatever your choice. It will take a little walk and you'd be on your own in quietness, alone with the fauna of the lakes.
Water density in the Small lake is the highest, so are the temperatures comparing with other waters - they areva few degrees higher than in Big lake, and even more than in open sea. It's good to know about this if you come in after season when open seas were well mixed with strong winds already, but the lakes are more closed an sheltered and mixing with sea water happens on slowlier time scale.
Equipment: Most of equipment you can get in Mljet NP, but if you don't want to head straight for shopping right after your arrival bring your own swimming suits and snorkelling glasseses. Boats and that - get by the park.
One of the options to get to the monestary island or if you just want some recreation, you can rent a kayak. Big & Small lake are very suitable for this sport as the water is very calm. It's also an alternative to the walking around the lake.
There are few points where you can get a kayak. One is at channel between Small & Big lake.
Mljet, being a nationa park, has very few cars and other motor vehicles. So, to explore the island (37km long, average 3km wide), the best way would be to rent a bicycle. In this way you can explore all the paths, not just the main road. And cycling in shades of trees is pleasant.
Equipment: There are several options to rent a bike. They provide also kid's bikes with extra wheels.