Life in Perth tends to revolve around the water, and that's one reason the Maritime Museum in the port area of Fremantle is a really popular place to visit. If you're into boats in any way, you'll be impressed - they have a huge hall with various boats literally hanging up in it, including Australia II, the yacht which won the America's Cup in 1983. Outside there's a submarine that you can walk (stoop!) through - it's fascinating although it made me feel pretty claustrophic and quite in awe of submariners who could actually stay in there while it's underwater, not safely ashore on dry land.
The museum is also about the many thousands of immigrants who arrived at Perth by ship from Europe and Asia. The Migrant Welcome Wall is something locals love to prowl around, looking for names of people they know who arrived by ship to Fremantle Harbour; my stepmother and her family rate a mention and I love to show their names off. There are great statues nearby representing newly-arrived migrants.
The entry price for the Maritime Museum is $10 for adults and $8 extra if you want to tour the submarine - I recommend it. While many of the other government-sponsored museums in Perth are free, this one is still worth the money - it's in a relatively new building and has got great exhibitions. It's open from 9.30am to 5pm every day.