By Slanted Light

Oh yes you'd need a car in order to explore better the area north of Perth. If I remember correctly, I could use public transport (from the central coach station in Perth) to go as far as Fremantle, City Beach (for this one also a standard bus was available, vs a coach), Cottesloe and Scarborough (which also has great coastline). But I don't remember being able to go further north, e.g. to North Beach or Sorrento, with public transport.

I also went on a 5-day coach tour once, out of Perth, to visit as far as Shell Beach and Coral Bay (both magnificent). The drive was quite long, and I preferred an organized tour because didn't fancy driving on my own on those isolated (and sometimes without mobile reception) highways. But the other side of the medal was, naturally, I wasn't free to stay and go in one place as long as I wanted... :)
In Perth don't miss the Botanic Gardens. Not only are they beautiful but they are in a panoramic, hilly location with great views!