Reef HQ - The Townsville Aquarium
This is the largest living coral reef aquarium in the world and a pretty cool way to spend a couple of hours.
You can get to the top from the bottom and back again in an under an hour so this isn't exactly the most testing of hikes in North Queensland. The view across Townsville across the sea to Magnetic Island is well worth the effort and the view inland isn't bad either. I'd 100% recommend getting up early and heading up here for sunrise.
A short walk from Rambutan The Strand is the best place to relax and enjoy the beach in Townsville. There is the potential for crocodiles to be in the water though and one was removed from this beach in 2016. So be careful if you do get in the water.
Great Barrier Reef
Townsville offers access to some of the most untouched parts of the Great Barrier Reef. You can find out more here.
Museum of Tropical Queensland
This natural history museum is right next to the aquarium and at any one time can feature anything from dinosaurs to lego. If you're unlucky to be get rain in Townsville I'd strongly recommend checking out this place and the aquarium in one go, maybe have lunch at Rambutan inbetween.
You should also consider visiting the Perc Tucker Art Gallery, taking a Jet Ski Tour and checking out The Queens Gardens.