Depending on where you are staying you can walk to most places, hire a bicycle, use the public transport or hire a car.
If you are staying at Cairns CBD you might be ok to walk to most places. Most of the hotel are 10 – 15 minutes to important locations such as Cairns Marina, Shopping Centres, Restaurants and Cafes. Also all Tour Companies pick up from the Cairns area.
If you are staying at Northern Beaches I might suggest you to hire a car. Even though most Tour Operators provide pick ups from the beaches you might want to have more flexibility while in Cairns and then you will be able to visit Cairns CBD whenever you want and try the many restaurants in town, go to the Night Markets, Shopping Centres etc. There are public transport available from the Northern Beaches to Cairns but they usually come every 1 hour and it could take over 45 minutes to get to the city depending on where you are staying.
If you are staying at Port Douglas you might also want to hire a car so you can drive to Cairns if you are interested in doing more tours in Cairns. Not a lot of tour companies provide pick us from Port Douglas as it is too far for them (1 hr drive from Cairns).