Though not really a tropical island, the indoor resort called "Tropical Islands" outside of Berlin makes for a great (albeit unusual) day trip. Located inside a former Soviet air hanger, the building has been converted into a biosphere with a tropical climate. Inside the resort there is an artificial beach, swimming pools, saunas & a spa, water slides, a small rainforest and even a hot air balloon!
To get there with public transporation, you can take a Regional Express train from central Berlin. Trains leave every hour toward Cottbus (depart at Brand-Niederlausitz). It takes about 50 minutes. From the train station, Tropical Islands offers a free shuttle bus for the 5-minute journey to the resort.
If you buy tickets at the door, admission to the Tropical World or the Tropical Sauna & Spa complex is 34.50€ for adults and 27€ for children age 6-14. A Kombi-Ticket is valid for both areas and costs 39.50€ for adults and 27€ for children.