I lived in Cairns for 15 years and am intimately familiar with good birding and other wildlife spots in the area. I could bombard you with info, but I'll try and be concise for personal faves:
Cairns environs: even a suburban walk around the city will yield a lot of bird species. Rainbow Lorikeet, Sulphur-crested Cockatoo, Magpie Lark, Pied Imperial Pigeon (in summer), Peaceful Dove, Figbird, White-bellied Cuckoo Shrike, Mistletoebird, Olive-backed Sunbird, several species of Honeyeater, Varied Triller, White-bellied Woodswallow, Bush Stone Curlew, Masked Lapwing and Metallic Starling, are all common urban birds, but there are many more. In winter, look along the esplanade for Pelicans, Jabiru, Osprey, White-bellied Sea Eagle, Brahminy Kite and various Terns. The variety of waders is likely to be not great, mind, though there may be some of the local Plover species. the adjoining mangroves may yield Mangrove Robin, Mangrove Kingfisher and Bar-shouldered Dove, but expect to be bitten to death. Asian House Gecko, Mourning Gecko and various small Skinks can be encountered in the city.
Wet Tropics Endemic bird species: Fortunately you can see all of these in a relatively small area west of Cairns. You really do need to head for the hills to the Atherton Tablelands to maximise your opportunities. Within a single day, but better spread over a few days, you can see all of them at such places as The Crater Lakes, Hypipamee, environs of Yungaburra, Julatten and Mt Lewis (these latter two are further north on the edge of the Mt Carbine Tablelands. Sometimes random spots can yield major rarities. The Sides Road hosted a small flock of Blue-faced Parrotfinches for several weeks - this can be a very hard species to find in some years. Fruiting trees may yield 3 species of Bowerbirds and a variety of Fruit Doves. Platypus are common on the Tablelands, and you may chance across one in rainforest creeks. Lumholtz's Tree Kangaroo is common around Yungaburra during the day.
Centenary Lakes: Located on the block on which I lived for 11 years and close to the city centre. This is a great place to see many of the above plus various Herons, Cormorants, Kingfishers, Darters, And with diligence, stuff like Lovely Wren and Large-billed Gerygone. In winter it may host Fairy Gerygone, Barred Cuckoo Shrike and other altitudinal migrants. I recorded 8 of Australia's 10 species of Kingfisher on this city block in the years I lived there. This is also an excellent spot for Brush Turkey and Scrub Fowl - they are easy here. It is also good for spotting Lace Monitors, and perhaps Snakes. Wood Frogs used to call from the lakes until a zealous contractor cleared invasive weeds a few years back. I didn't hear them after that... Graceful Tree Frog, White-lipped Tree Frog, Spotted Marsh Frog and both species of Sedge Frog can be common some years and sometimes call in winter. Little Red Tree Frog often calls near the mangroves at night.
The track up Mt Whifield nearby may be good if you head up into the remoter parts, but is infested with noisy dayglo joggers further down. Good to do a full circuit just to experience it, really. You may encounter Agile Wallaby, Musky Rat Kangaroo and Red-legged Pademelon - even Echidna; perhaps also mixed flocks of Red-browed Firetails and Silvereyes, as well as Topknot Pigeon and Cicadabird. If you hang about the vicinity of Collins Avenue (where I used to live) after dark, you may see hunting Rufous Owl, Striped Possum and Bandicoots. You will certainly see Spectacled Flying Fox over the city in the evenings and hear their noisy banter after dark.
Vine scrub: Emerald Creek is good, but there are other patches of it here and there. Necessary if you want the very localised White-browed Robin, especially.
Tropical Savannah: The vicinity of Mareeba is a good area to explore. You could pick up Australian Bustard, Australian Pratincole, Red-tailed Black Cockatoo, Pale-headed Rosella, and various Roos just driving along. Granite Gorge Nature Park west of Mareeba is a good place to guarantee Squatter Pigeon, and often has a roosting Frogmouth or Owl which the park owners usually have staked out. They also have very tame, and very overweight, Mareeba Rock Wallabies in a colony there. The area around Mount Molloy is the best for Great Bowerbird, but like so many places here, just explore and you will see good birds.
Everyone goes to the Daintree it seems. Good if you want to see crocs - you can see them readily from a Daintree cruise - but missable if you are after other stuff. Cassowary House no longer exists now that Phil and Sue have retired. Phil posts on here sometimes as Sicklebill. Local birders that you encounter in Cairns can be very helpful, so pick their brains for latest info. That's how I was able to find the Parrotfinches and a nearby nest of Square-tailed Kite one year.
Michaelmas Cay: not sure how good this is in winter, but the island is heaving with Sooty Tern, Brown Booby and Brown Noddy during breeding season. A day trip would cover it, and you get to go to the Great Barrier Reef (or a rather knackered part of it) in the bargain.
Speaking of Cassowaries: as has been said up the page, these can be very gettable at Etty Bay, south of Cairns, but I have also seen them lots at Hypipamee.
There are many other locales in the region worth seeing, but those would be my picks!
