I rarely post here but read this and thought I'd chip in. We flew to Perth last April for a three week jolly with the kids, originally to have a look at possible relocation but things changed between booking and flights, so it became a holiday only.
We also flew direct from Heathrow, worth it if you can.
Ten days in the region is long enough to get in all the sights & depending on your budget I would recommend getting a car. Everywhere is literally half a hour away.
We did a lot of things that were family friendly and as a solo traveller may not float your boat, we spent three days in the same water park for example.
Rottnest Island is something you should really look into, the place is absolutely beautiful and a nice ride around, I saw some scuba divers there, so could be worth checking out.
Freemantle is a good day out too, it's got a cool af market & plenty of places to eat & drink, some with ocean views. There's a speed boat in freeo that takes you out into the shark infested water and chucks you around for a bit if that's what you into. Actually you could probably get two days out of freemantle with the museum's as well.
If you looking for a bit of wildlife check out caversham wildlife park instead of the city zoo. It's cheaper and much larger.
The city in my opinion, bar kings park, is the same as any city in the world. We did a tour bus trip that takes you round the city and through kings park, where you get off and pick it up later on. This I would recommend. The park is immense.
Do you like wine & chocolate ? There's plenty of wine farm tours that will keep you happy and again depending on you budget you could go up river.
The thing that struck me about Perth is the beaches. North and South of the city the beaches are another level. There's a foot/bike path that goes north of the city up to burns beach. If your able to get a bike that would be a fantastic ride.