Whilst Rarotonga and Aitutaki both have some fun adventure tours, the islands are rather too commercialised and too small in size to offer any real adventures. The only islands where you'll experience true adventure in the Cook Islands are on the eastern islands of Atiu, Mangaia and Mauke but here accommodation is limited to just one of two small guesthouses and provisions can be unreliable. Extreme travellers may want to consider taking the monthly cargo boat to the utterly remote Northern Atolls where government guesthouses provide the only beds for outsiders.
Atiu is good island to explore, mostly because you're unlikely to come across too many other travellers, and quite possibly none at all. There are caves to explore on the southern end of the island and the central plateau is a place popular for farming coffee and local vegetables and a good place to meet the locals in their village environment.
On Aitutaki, the main island is still very undeveloped and by hiring a motorbike its easy to explore and get a sense of outer island life. The southern end of Aitutaki is the least developed and best for exploring the bush, whilst east Aitutaki has some nice villages to visit. If you dream of visiting an uninhabited island, you'll either end up sharing your dream with a handful of other tourists. To escape the crowds, take a private cruise of the lagoon or stay at Akaiami island where an exclusive lodge has opened catering to just a handful of guests at a time.
The best region of Rarotonga to get away from the crowds, to explore the hinterland and have a sense of living amongst the local people is along the east coast of the island from Matavera Village to Avarua Town. Heads into the foothills and explore the rainforest and plantations. There's several excellent hikes - the Cross Island Track and the trail to Rimeau Peak.