Plenty of beautiful suburbs in Perth. Look at Mt Lawley, Inglewood, North Perth, East Perth. Close to the city and the river you have South Perth, Como, Applecross, Ardross, East Fremantle, Fremantle.. Subiaco is a very popular suburb with flight attendants and so is Cottesloe (on the beach). If you would really like to go upmarket maybe Nedlands and Claremont would be good. Hope this will get you started.
Good luck with your search and hope you have a great time in Perth.