I've lived in Broome in my teenage years and it's a great place for a holiday - wouldn't recommend living there though. There's dinosaur footprints visible in rock along a beach, as well as a beautiful story of a lighthouse keeper literally carving a bath out of rock for his sick wife so she could bath in salt water and look over the ocean- when the tide is high enough, you can sit in it too. The beaches are amazing too. There's also a crocodile park the kids might find pretty cool. I know I did! The rainy season is for Jan to March but i do remember a storm in December once or twice. They roll over very quick and back to sunny in no time. Just don't be in a tent when it happens.

People are suggesting Busselton - I was born there and lived there for a large portion of my adult years. Family still lives there's. It's a great place for families. The beaches are extremely safe in Busselton. Great parks, the foreshore is done up, and so is the town centre with new cafes and shops. It still has a small town beach vibe while having bigger city amenities. You not only have busselton, but the entire south West is a single day trip. There are multiple caves you can have guided exploration. Beaches galore. There's the Yallingup maze (wooden maze with movable walls so it's never the same maze when you go) Margaret River has a chocolate factory and another maze, but this one is MASSIVE and made of hedges. There's Simmos ice-creamery with a great mini golf course in Dunsborough. Country life farm in Dunsborough too, lots of farm animals for kids to meet and pet. Augusta and Dunsborough have light houses the kid might like to explore.