Amami Airport (IATA: ASJ)  is located in the east of the island. Japan Airlines (JAL) , Vanilla Air , Japan Air Commuter (JAC)  and Ryukyu Air Commuter (RAC)  operate daily flights to/from Amami Airport for a number of domestic destinations.
A southbound (for Tokunoshima, Okinoerabujima, Yoronjima, Motobu and Okinawa/Naha) and a northbound (for Kagoshima) ferry operating daily by Marixline  and A"Line . Normally a southbound ferry arrives at Naze Port at 5am and departs for Tokunoshima at 5:50am, a northbound ferry arrives at 8:30pm and departs for Kagoshima at 9:20pm, with occasional delay due to bad weather or loading cargos.
At http://shimabus.jp/ office you can buy a bus pass for 1/2/3 days for 2,000/3,000/4,000 yen respectively (a roundtrip from Naze city to either airport or Koniya port costs more than 2,000 yen anyway).
To get to Kakeroma Island, take a bus to Koniya port and get a ferry to either one of the two ports on Kakeroma.
Some hotels have their own bicycles to rent for about 100 yen per 2 hours.
There is a Habu snake center in Naze city, with a small serpentarium in the basement.
K'NA tourist center (http://www.amami.com/) offers several itineraries, such as canoe tour through mangrove forest, a day tour in prehistoric forest with tree ferns, and an after dusk photo-safari car tour in the woods where you can encounter local snakes, frogs, mice, insects and the famous black rabbit almost identical to fossil rabbits from a different age (make sure to take a good camera!).
Ohama beach is not far from Naze city (~30 min by bike); there is a small deigo tree grove blossoming in late April, a small marine life museum and a nice Thalasso SPA with a bubble bath on a balcony with a sea view (good for watching sunset).
Tomori beach (near the airport) is probably the most beautiful one.
On Kakeroma island, there is a famous deigo tree alley (Deigo Namiki) blossoming in May/June.
Visitors should be aware of the Habu snake. A venomous pit viper that inhabits roughly 70% of the island.
When diving, swimming or snorkeling at the various beaches watch out for sea urchins lying on the sea floor.