Rhodes is not one place, it's three or four stacked on top of each other. The Knights Hospitaller took the town in 1310 and held it until January 1523, when they marched out after Suleiman the Magnificent's second siege. The Ottomans stayed until the Italians arrived in 1912, the Germans took over in 1943, and the island only became Greek in 1948. All of it is still visible inside the walls of the Medieval City of Rhodes, a UNESCO site where a Gothic hospital, an Ottoman mosque and a Sephardic synagogue stand a few minutes apart.
Outside the walls the island is about 1,400 km2 and the good things are far apart: a whitewashed village under an ancient acropolis in the south-east, Doric ruins and Crusader castles on the west coast, a tiny island of sponge divers an hour or so away by boat. That's why five days works here and three doesn't. October is a good month to do it: the sea is still around 24°C, the crowds have thinned and the big sites keep their summer hours until 31 October.
The plan keeps two days in Rhodes Town and spends the other three on the road or on the water. Day one is the Old Town, day two the new town, Kallithea and Monte Smith, day three Symi by boat, day four Lindos and the east coast, day five the west coast. Start on a Wednesday: the Old Town's wall walk runs only Monday to Friday and only until 31 October, and from 1 November nearly every state-run site is closed on Tuesdays, which this order never touches. The rest of Greece is a ferry or a short flight away, but Rhodes alone can fill the week.