The Botanic gardens are a beatiful spot to spend a day in lush Queensland rainforest, right in Cairns. There are spots for picnics as well as really interesting displays about native and local flora and fauna, as well as more exotic species from the surrounding tropics.
I recommend getting a taxi/uber to Lily Creek bridge and walking up the hill from there. You can visit the botanic gardens and make your way into the cafe district in Edge Hill for some lunch. A great, cheap and hidden locals day in cairns.