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.
This is one of the most beautiful natural places on the island of Rhodes. In English, it's called Seven Springs, but in Greek it's called Epta Piges. These are natural springs that are open to the public. It is a particularly romantic place and great for couples, especially those on honeymoons. The springs are located in the woods, and have amazing scenery. It's the perfect place for a picnic and to spend the afternoon. I highly recommend that everyone visit!
Pefkos is an absolutely amazing destination to go to on the island! It has a great beach and wonderful restaurants and coffee shops in the area. The beach is not overcrowded and I personally think its a great place to go with family and kids. There aren't too many organized activities on the beach but it is great for swimming. Here is a great areal view of the beach.
I am very happy to see that Greece is investing in preserving its architecture and mediaeval cities. Rhodes has so much beauty, that I would hate to see it go away overtime because of environmental or other damages and erosion. Their plan is to create three main sectors of the city which will be the monumental, residential, and commercial areas. I'm very curious to see how this project will turn out and how the funds will be allocated. Have you heard anything about this?