It's great. Lokrum beach is a five minute boat ride from the old town harbour - it's all rock flats so it's easy to find a spot to lie down. The sea's gorgeous, very warm and clear, and there are high cliffs you can go jumping off (but if you go off the highest ones, make sure you hit straight or you'll bust a rib like I did :P). Also, there are wild peacocks that may come and say hi! On the mainland, if you walk around the Lapad Peninsula and through the campsite, there's a nude beach called Cava. We almost didn't find it the first time, as it's not always signposted, but you keep going past the main beach (Copacabana) and then take another right, which leads you down to Cava beach. The near half is clothed, but once you hop across the big rock with 'FKK' painted on it in big red letters, you're good to go. It's warm enough to sleep on the beach, and Cava is a lovely spot to sit and watch the sun go down behind the Elafiti Islands.
Other than that, I've been to Split, Hvar (Jelsa, Stari Grad and Hvar Town), Sibenik, Trogir (although only for a day), and Zlarin. Got plenty more places to explore yet, too! Let me know if you're planning to go to any of them and I'll see if I can tell you anything useful :)