The cheapest way to make a boat tour to the islands outside Gothenburg: Take the ferries that connect the islands with the main land. They are part of the cities public transport system and that means that any ticket for bus or train is valid for those boats as well.
Boats leave from Saltholmen in the far west of Gothenburg and allow you to sail to many islands of the southern atchipelago, hop on and off or just visit one island and return to Gothenburg. Timetables and a little map are availible at the terminal in Saltholmen.
A visit to Brännö