Here people usually grab a drink or several, watch the sunset and stay until late. Be sure to get there early though for seats as space is limited and the popularity of this bar is increasing.
To get a view of the city walls from the water, rent a kayak just outside Pile Gate at Kolorina. Don’t fancy kayaking, then just soak up the sun at the beach itself (also Kolorina). Another great spot for swimming is just along the outside of the city walls, near Poporela, the 19th century pier by the harbor. From this spot, if you keep walking along the outside of the walls, away from the harbor, you will see people swimming and sun bathing on the rocks nearby.
Dubrovnik old town may be small but it offers a great variety of experiences. It is only getting more popular as a tourist destionatoin so I would recommend putting it high on your list of places to see.
Also: This is a popular destination for Game of Thrones fans, since ‘Kings Landing’ is modelled after Dubrovnik. For all enthusiasts, there are tours that take you to each filming location.