When you enter the Old Town through either the Pile or Ploče main gates, you'll find yourself on Stradun, which is the cobbled main street. A great place to start is by walking the 2km-long city wall, which has a spectacular view overlooking Dubrovnik's famous red roofs.
There's one access point just inside the Pile gate – follow the stairs up to the ticket office and onto the wall. You'll need 2-3 hours to do the walk and take in the sights of the Old Town and the sparkling sea and islands beyond.
Back at street level, explore the maze of steep streets that rise up on both sides of Stradun. Finally, make sure you stroll down to the old port – it has a wonderful atmosphere and lovely views along the coastline towards Cavtat.