Most travellers coming to Borneo on package holidays to either Kota Kinabalu or Kuching will have airport transfers already included in the price of their holiday, although if you have not arranged them, then you can get a taxi to your hotel from either airport on arrival, although this can be more expensive - particularly if your hotel is quite remote.
Between Sabah and Sarawak, the best way of getting around is by air, and this also applies to longer distances within the two territories as well, because roads are not always very good off the beaten track and the standard of driving locally is quite poor.
The best way of making short trips in Borneo is by using the bus services and private taxis that can be picked up from most hotels and in the main towns, although the buses can be unpredictable and operate to strange complicated timetables. You should try to agree a price in advance with a cab driver rather than using the meter - particularly on longer journeys.
Many holidays to Borneo are undertaken as a guided tour, where meals and all transport around the island are included, and this option, while not offering much in the way of freedom does ensure that you get the chance to see all the main sights during your stay.