We visited Cairns in January a few years ago and had no problem. As others said, it was very hot and humid, and we did barely dodge a cyclone (which stayed north). The rain is more likely to be spotty, occasional downpours than constant showers.
Being newly certified divers will be no problem. There are of plenty of relatively shallow dives that are wonderful. As elsewhere, some of the best diving is in very shallow water - as little as 10 feet. The colors are better, and the bottom time can't be beat!
I did a 4 day trip aboard Taka II that went to Cod Hole, and several other sites before heading to the Coral Sea. My non-diving wife and son went on a day trip snorkeling on the reef and called it a "religious experience".
Australia is fabulous - beautiful, uncrowded, friendly, exotic yet familiar. We can't wait to get back!