"Where the red dirt meets the sea", the slogan of the town could not be more appropriate. Don't expect a town that lives up to the proportions of our European towns. Like lots of small Australian towns, I found it uninteresting but it is the biggest for thousands of kilometres around.
Broome used to be at the centre of the world pearl industry. Nowadays, you can buy pearls there, visit a pearl farm, or even visit the pearl museum. Cable Beach is a favourite spot, to be seen at sunset. You can even have a ride on a camel's back along the beach... We keep ourselves busy with what we can in the North of Australia!
Another possible activity: the seaplane over the "horizontal waterfalls" in Talbot Bay, impressive rock formations. I wasn't able to do it but the view and the contrasts of colours are amazing.