On the island we would recommend you to visit nice sandy beach of Saplunara, gulf of Sutmiholjska and very special small villages of Prozura and Kozarica. You should not miss the famous Ulysses' cave too.
To the one-day excursion off the island you can travel by an organized speed boat or at your own. You can visit the beauty and history of the world known city of Dubrovnik but there are other possibilities like visiting the neighboring island of Korcula - you can see the famous house of Marco Polo there.