well, Perth as a city is quite small and there isn't a whole lot to do around the city centre itself. The beaches are where the action is - Cottesloe Beach is probably the nicest but they're all good really. The beaches are very close to the city. Wind and kite surfing are very popular.
Fremantle is a lovely little town (a suburb of Perth really at this stage) and is worth a couple of visits - a lot of interesting convict history, if thats your thing.
Rottnest Island (about 20kms off the coast) is a great spot for a day trip - hire bikes and take them on the ferry (leaves from Fremantle). You can cycle right around the island is a few hours and there's lots of nice beaches and coves to stop off and have a dip.
There are wildlife parks, zoos, waterworld parks etc around Perth - I lived there for almost 2 years but never visited them so I can't comment.
Any other trips out of Perth typically take a few hours drive. Margaret River is the most popular - a big wine region and also a good spot for surfing.
Distances are long since the state is vast.
Freemantle is a bit overrated.
Yanchep to the north is couple of hours drive and offers nice park with loads of cockatoos, koalas, kangaroos etc.
Nice to wander aorund for a few hours.
You can go further north upto the Pinnacles but you are talking a full day to go up and come back. There isn't anything there apart from the rocks.
Margaret River is more than a few hours drive, beware.
Margaret River is nice and you can visit some wineries, we visited Luewin Estates, the one that hosts the classical convcerts. It is a bit pretentious so then we visited small family run one which was way more relaxed and down to earth.
Further south you can do a fantastic drive through the very tall tingle and yarri tree forests, the roads go right between these colossal tree and you can visit the Valley of the Giants and it's tree top walk near Walpole.
You could allocate 4/5 days to do this trip. You can visit the Southern Ocean but it is cooler than the Indian Ocean for swimming.
Do not go for the drive upto Monkey Mia to see the dolphins. It is featureless road according to my missus, the most boring trek in the world. Personally I thought it was interesting counting the dead roos along the road.