+1 for Perth
Several nice suburbs already mentioned,Subiaco,Fremantle(freo),Leederville,& Nedlands/Peppermint Grove if your wadded.
Personally I found Perth(city centre) easy to get round on the bike,we used to live in Wembley which was about 10/mins bike ride.
There are some crazies(hoons) in cars on the roads though.
Road cycling is a lot more popular there with a few of the Bike shops(The Bicycle Entreprenuer)doing group rides if your interested.
For Mtb head to the “Hills” literally around Kalamunda ,plenty of trails to ride and some great scenery.
And a plug for my old workplace