The Dalmatian Coast is famous for its calm and blue water, which is perfect for swimming and its pebble beaches.
While there are no beaches in Trogir's city center there are a number of beautiful options within a short walk or drive. Medena Beach is in Seget Donji, which is a village located some four kilometers from Trogir. It is a pretty pebble beach that can get crowded in the summer, but is still fun.
Also check out Ciovo Island, which is across the bridge from the old town, and also hosts quite a few beaches.