The island capital is the large and busy Vathy (or Samos). The waterfront is a string of bars and restaurants, and although very pleasant, the nicest part of town is the upper town of Ano Vathy. There you'll see red-tiled 19th Century houses on the hillside. At the northern end of town is a small string of beaches, the best of which is Gagou Beach, about a km from the town center.
Vathy has an interesting archaeological museum with the largest standing kouros (male statue of the Archaic period) anywhere. There are other fine sculptures and pieces worth seeing.
Cobble-paved Pythagorio on the southeast coast of the island is rather upscale. It's beautiful with red hibiscus and pink oleander lining the streets. Many of the ancient sites on Samos are accessible from Pythagorio. Boats from Patmos and other points south all dock here.
Pythagorio is a World Hertiage listed city because of the ancient city dating from the Neolithic era. The best remains are from Polycrates time and can be seen just beyond the jetty.
From the town center you can walk up the Moni Panagias Spilianis (Monastery of the Virgin of the Grotto). The Roman baths are nearby. The remains of the Castle of Lykourgos Logothetis are worth seeing, too. Other points of interest in Pythagorio are the Pythagorio Museum, the ruins of Aphrodite, and the remarkable Evpalinos Tunnel. The statue of Pythagoras is near the water.
Near Pythagorio, The Ireon or the Sacred Way was once flanked by thousands of statues that led from the city to the Ireon. This is the legendary birthplace of Hera. Her sanctuary was built in the 6th Century BC. There's been a temple there ever since. The Ireon was four times the size of the Parthenon. You can still see its size, although only one column remains after centuries of plundering and earthquakes.