Lokrum Island sits just offshore from the old town area of Dubrovnik. There are ferries to the island about every 30 minutes or so (maybe more often in the summer). The ferry ride takes about 15 minutes and you get some great views of the old harbor in the process. We didn't expect a whole lot out of our trip to Lokrum but it turned out to be a great half day trip and we would definitely return.
That little pier in this picture is the harbor for Lokrum. That should set your expectations fairly low although it turns out that there are up to three restaurants on the island (all with bars) and some sort of ice cream / snack stand. People swim from the rocks around the harbor and the water is very clear. This is not great swimming for children but the rocks are great sunning spots. Expect toplessness here.
The island is infested with peacocks. They are everywhere and they want your food. If you aren't eating food they will go back to ignoring you. They are most common around the harbor and the restaurant areas.
The west side of the island faces away from the Croatian mainland. There are a handful of ruins over here, good views of the outer islands and one or more nudist beaches. There is also a good docking harbor for canoes and kayaks where you don't have to contend with the ferry shuttles.
There's a sinkhole on the south end of the island known as Mrtvo More (the dead sea) which is one of the highlights. This is better swimming - possibly all around. It is well protected, although there is clearly some interaction with the nearby sea as waves seem to spontaneously appear. The name comes from the salinity which is not as high as the Dead Sea but it is concentrated.
The bottom is rocky and uneven but the water is very clear and tends to be warmer than the sea water temperature. One of the island's cafe-restaurant-bar type things is located just outside the main access to the sinkhole.
Rabbits are the other critter of choice on the island (besides peacocks). They are large and unafraid of humans. The island also has a botanical garden and a "hill" on the north side. The hill is not all that daunting. Supposedly, Richard III (Lionheart) shipwrecked here on his return from the crusades. It's unclear if there's any evidence behind that or if it just makes a nice story for the brochure.
There was definitely a Benedictine Monastery on the island because it mostly still stands. There's even a hedge maze amongst the cloisters. Our son found this particularly charming as it was also swarming with rabbits. The monastery is maybe a five minute walk from the harbor. The sinkhole is another 5 minutes past that. The entire island can be covered in a few hours. The flatter (south) side of the island can be seen in less than that although it's worth spending some time swimming in either the sea or the sinkhole.