From Cairns Airport to the city centre:
Most hotels offer free pick up at the airport for customers staying more than 1 night with them. Check with you hotel if they have that service. Otherwise, take the public shuttle that leaves from the airport (the orange booth). It will drop you at the entrance to your hotel for 15 AU$ (10 minutes). A taxi costs 25 AU$, so if you're a party of 2 or more, it makes sense to take one.