Only 40km from the Montenegrin border lies the 1300-year-old jewel of Dubrovnik, whose appeal rests in the "Stari Grad" or old town. Within city walls lies a pedestrianised haven of marble-paved (slippery in the rain!) squares, steep cobbled streets, monasteries and convents, churches, palaces, fountains and museums, which are all hewn from the same honey-coloured stone. Motorists, users of public transport and day-trippers all arrive at one of three car parks and enter through the city's old gates. Though heavily and pointlessly shelled in 1991, Dubrovnik has since benefited (and well used) UNESCO reconstruction aid money.
The town is bisected by its main pedestrian promenade (the "Stradun"), running from the Pile Gate to the clock tower at the other end of town (and beyond to the new harbour). Just inside the Pile Gate, you can't miss the unstriking, but undeniably enormous, circular fountain, opposite which is the Franciscan Monastery which houses Europe's alleged second oldest operative pharmacy (apparently in continuous use since 1391). At the other end of the Stradun stands St Blaise's Church and the 1441 gothic Rector's Palace, now a museum with furnished rooms, baroque paintings and historical exhibits. In the next square along is the town's market.
The highlight of a trip to Dubrovnik is a walk along the 2km of its 25m-high city walls, dating from C13-16. From here, you have an unparalleled and ever-changing view across Dubrovnik's terracotta-tiled rooftops, flower-decked terraces and balconies, the harbour (still being constructed but the bobbing fishing boats make a pretty picture), St John's Fort (outside the walls and itself worthy of a visit -- more steps I'm afraid - €1.5) and the many spires and towers of the city as well as the glittering Adriatic.
If you do decide to stay overnight, remember accommodation inside the city walls is limited and therefore at a premium -- the choice is either to stay in a hotel in Lapad down the coast (usually bookings have to be for three or more days), splash out for a hotel in the town or accept an offer of a private room from one of the women waiting at the ferry terminal or bus station (or you can book one through the internet but will pay an agency fee of up to €30 for the privilege).
I can't resist one last recommendation -- the small cafe/bar next to the new harbour -- the most fantastic garlic squid or vast pan of whitebait for 35Kn (c£3/€5) per portion and excellent salad.