During the time of the Crusades in the Middle Ages, the Knights of Saint John had their headquarters on the island of Rhodes and there are many structures associated with those headquarters. There's a hospital, the Palace of the Grandmaster, etc. Besides that, the island has ancient history, as well. The Acropolis on Lindos is one of the ancient places that are cool to visit.
Taking a day trip is the easiest way to see many of the islands in Greece. Symi is a pretty small island, so I don't recommend spending many days there. What you can do is take a short ferry to the island and spend it walking around, exploring and heading to the beach. If you're lucky, you'll be able to check out the annual music festival which features artists from around Greece.
Many people don't know that Rhodes' Italian influence is pretty recent and not from Roman times. The last time Italians set foot in Rhodes was in 1943 after World War 1. It was only until 1947 that Greek became a part of the modern Greek state. Italians really contributed to the development of the island and to the culture and architecture. This is one of the reasons why Rhodes is so unique and has a pretty different vibe than some of the other islands.