I would start by taking one of the Waxholm ferries. If you do it before the end of April (I think) it won't cost you anything if you have a 30-day, or longer, SL ticket. You can start by going from Nybrokajen in the middle of Stockholm to Vaxholm, Ljusterö or Sandhamn and back. Waxholm ferries are a cheap and comfortable way to see the archipelago.
Bring picknick so you don't have to pay for the expensive and not very interesting food on the boats. Sunglasses, sunscreen, and a windbreaker are other good to have accessories.