The island Lokrum has a large monastery, mostly now ruins, gardens with peacocks walking round. I believe there is bike rental there too. Swimming is off the rugged rocks, with ladders to help you in or out of the water.
But the main draw to the island for us is the cheap boat trip, the shaded café bars and a large deep lagoon. It is where you sit and watch ‘mad people‘ jumping and diving from various ledges off the cliff face, and then realize they are members of the family! Swimming in the warm lagoon is also fun.
The facilities on the island aren’t extensive but there is a couple of café bars, one selling snacks and clean public conveniences and showers. Ideally take a picnic.
Tip: The ferry boat runs to a time table but it ran half an hour late when we last went, and we also hadn’t realized that on the way home we went via Cavtat and Milini. It was lovely sailing as the sun was setting, but do check on return arrival times so you can plan. We will leave a seasonal timetable with the excursion file in your accommodation.