Lampedusa is small enough that you don't strictly need transport — the town and its nearest beaches are easily walkable. For the more remote beaches and the nature reserve, a scooter is the ideal solution: rental shops in town are plentiful and cheap, and the roads are easy.
Boat trips are well worth organising. Most days in summer you can join a shared boat excursion around the island, stopping at sea caves and remote coves that you cannot reach any other way. Ask at the harbour or through your accommodation. If you want more flexibility, small boats are available to rent by the hour or day.
There are no buses to speak of. Taxis exist but are informal — ask your hotel to call one if you need it.