Visitors to Stockholm can connect with the beauty of the thousands of islands via convenient day or overnight trips from Stockholm's inner harbor or from Vaxholm (itself a popular afternoon destination by boat). Two of the biggest companies running boats out into the islands are Waxholmbolaget and Stromma and the most popular area to explore is central skärgården (The blue on the map here). Each island has a different feel and different set of amenities: Some are virtually unpopulated; Some are resort like; Some have only a small shop or restaurant and camping to offer; Some are lush and green, while others are mainly made up of the bald rocks that are part of the people here. In my case, I wanted to get out, spend the full day on the boat, and relax, so I picked Sandhamn as my destination.
Getting out to Sandhamn takes a little more than 2 hours. If you are looking for something close, Grinda offers a bit more of the island feel than Vaxholm, but is still doable in an afternoon. For overnight stays, the hostel/camping on Finnhamn offers an excellent experience (complete with the option to rent a sauna). On arrival, you'll find yourself in a small downtown area where the hotel and restaurants are located.
Outside of this, the island is a very Swedish mix of nature and development. This is a community mainly of summer homes and people drawn here by the promise of a beach. The island does, however, have a permanent community and the local government is active in promoting Sandhamn as a year-round destination. The beach itself is a short walk to the other side of the island (Everything on the island can be reached within a 15-20 minute walk). It is a beautiful little beach and immensely popular. No matter where you go, it is hard to go wrong with a trip out to the islands on a summer day.