the best dive company in Cairns, hands down is Tusa Dive . The staff, the service, the boat, the dives are all top notch! In all my dives, I haven't been waited on hand and foot, like TUSA did. The staff was amazing answering questions via e-mail beforehand and ensuring I was doing well from the moment I stepped on the boat, to the time I disembarked. The divemasters all know what they're doing and made sure we had a good time. TUSA has the fastest boat of the dive operators, so they dive the outer reef, which very few companies dive, so you will not have tons of boats scaring off sealife. The spread they prepare for lunch is also the best I have had on a dive boat.
I know they have a liveaboard, but I can't speak to it. But if you are looking for day trips, definitely book TUSA.