Getting around the Cook Islands on your gap year is easy. In essence you can either fly or catch a boat (or walk; or cycle; or get a bus - ok, so there are lots of ways to get around the Cook Islands but they're all really easy...)
The 'Round the Island Bus' operates regular services around Rarotonga from Monday to Saturday. The bus drivers will be happy to stop to pick you up anywhere along the way and when you want to jump off just say so.
The international airport in the Cook Islands is on Rarotonga (RAR) and is 3km west of Avarua.
Domestic inter-island service is provided by Air Rarotonga. Although you can book flights through Air New Zealand; it's usually cheaper to do so directly with Air Rarotonga. This has become much easier in the past few years as they now offer online booking.
If you like roughing it you can visit all inhabited islands by inter-island freighters, but these can be weeks apart or worse if you want to get yo the really remote islands. Details of services are published in local island newspapers.
There are no generally scheduled boat or ferry services between the inhabited islands.
There are two uninhabited islands - Takutea and Manuae. The only easy way for a visitor to get to Takutea is on the research vessel Bounty Bay operated by Rarotonga-based Pacific Expeditions, which has special permission to run occasional eco tours.