The island of Utö is in the Stockholm Archipelago South East of Stockholm. We suggest you fly to Stockholm.
From Stockholm Arlanda Airport you can take the Arlanda Express train to Stockholm City Centre.
From the Stockholm Central Station you can take a subway, bus and ferry to Utö. Use the travel planner in this link and put in Stockholm C as your departing station and Utö Gruvbryggan as your arrival station.
If you arrive by car there is a big parking lot at Årsta Brygga where the ferry to Utö, Gruvbryggan leaves from.