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.
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.
Rhodes has some of the most beautiful villages from any of the Dodecanese islands in my opinion. I love to roam around, take pictures and stumble upon quaint restaurants with amazing locals. This is just my personal list, and of course I'm sure there are many more villages that are equally, if not more beautiful.