Like so many things in life - it depends. It depends what you're looking for.
Perth as a city cowers an enormous area due all the new suburbs they keep building. You don't want to end up in any of those! Some of the older suburbs nearer the CBD (the only place with skyscrapers) are nice in terms of culture, food, and drink. I'd put Mt Lawley on top of that list.
Nearby is Northbridge, which you probably want to avoid. Unless you come for the bindge-drinking teenagers and pill-heads going clubbing. They've got great cheap Asian food there though!
If you're in Australia you probably want the beach. And luckily Perth has some of the better beaches. But the beach is a good 20 minute drive from the city. The nicest ones near the city are Cottesloe, City Beach, and Floreat. At Cotteslough you might be able to find some accommodation, but it depends what time of the year you are going. It's probably pretty dead in winter there.
About Perth: It's huge and people drive everywhere. There are buses, but they aren't worth much. The metro/train is great though, if you are lucky enough to need to go somewhere along it's north-south line. Cottesloe being one of those places.
And finally - Fremantle, the best of all of the above. A port-town swallowed by Perth's expanding, but has kept it's own vibe. More of a hippie vibe if you look in the right places, plus good beaches, nature reserves, old houses, the river, bars and restaurants.
-- wow, just realised I went on a bit of a rant there, possibly without even properly answering your question. For 2-3 months I'd have a look at http://www.gumtree.com.au/ or https://www.couchsurfing.com/places/Oceania/Australia/Perth.