The most common way to get to Zamami is by high speed ferry ("Queen Zamami 3") from Tomari port in Naha city on Okinawa Island. The crossing takes 50 minutes and there are usually two departures each day (¥3,175 one-way, ¥5,970 round trip). There is also one trip each day with a bigger ferry ("Ferry Zamami"). For departure times check the timetable  (in Japanese). Note that the high speed ferry is canceled in rough weather (waves higher than 3m).
There is also a boat to Akajima that runs four times a day. (¥300)
Zamami is a small island; it's possible to walk around it in a day. Other options are to rent a bicycle or car.
There is also a bus service that runs between the port and the campground to the West and Furuzamami beach to the East.
The Humpback whales return to to give birth to their young, then nurture them in the waters around Zamami from January to March every year. There are many different whale watching tours available from Zamami port during that time.
There are nice waters around the island so SCUBA diving is the most popular activity. Other options are snorkeling and sea kayaking.
The island has many dive charter companies, diving is there main tourist attraction.
There is a campground about a 15 minute walk from the ferry port on the beach with toilet, shower and bbq facilities. They also have some camping equipment to rent. The campground is in a grove of shady pine trees right on the beach.
There are a number of diving and snorkeling points around the island but most snorkel at Furuzamami beach and the campground beach.
There are a number of geocaches on the island, most of them at the lookout points. http://www.geocaching.com
Zamami Festival is held usually at the end of the summer with Eisa dancers, dragon boat racing, entertainment, and fireworks.
The annual Sabani Race from Furuzamami beach on Zamami Island to Ginowan in Okinawa runs the last weekend in June or the first weekend in July most years. The sabani race is usually run in conjunction with the yacht race from Ginowan Marina to Zamami port with the yacht race being held on Saturday and the Sabani race held on Sunday.
There is an excellent guide in English written by the English teacher on the island 
Take the boat to Akajima or go on a tour boat.