You can book a water taxi or rent a kayak to get to the island. You'll need to pay admission upon arrival.
I found it easier to book the transfer and tickets through the Reserve so it's all together.
You can buy tickets online or at the harbor. The East Gate is closest to the harbor. Walk forward from the east Gate. You'll reach a dead end with an arched doorway to the left. The ticket office will be on your right just after the restaurant.
Tip: Buy tickets online to save 10%.
The transfer to the island departs on the hour and half hour. The returning boats depart at :15 and :45 after the hour. Last transfer back is 5:45 pm. The sailing takes about 15 minutes. Be sure to double check the schedule because these times can and do change based on the season.
Tip: Boarding starts about 20-30 minutes before departure at the harbor. Grab a spot on the back (aft) of the boat either downstairs or upstairs for great views and pictures of the harbor as you sail toward Lokum.
The pink building is the visitor center. Head in to pick up a printed map or to buy the audio tour.
Be sure to bring your swimsuit and a towel to enjoy the several swimming areas. (Or just a towel if you are headed to the nude swimming area.)
Tip: Bring water shoes and/or walking sticks if you aren't surefooted.
I recommend heading to Fort Royal first since that's the highest climb.
The Monastery is where you'll find the Games of Thrones exhibit and Iron throne.
On the way back, the boat arrives 5 minutes prior to departure. People queue up so you may want to arrive a little earlier.