I moved to Perth with my wife and 3 year old daughter 4 years ago, but not from the UK so can't make any direct comparisons.
Perth will certainly give you the relaxed lifestyle you seek, and is a very outdoorsy place. It's a great place to bring up kids - safe and healthy, if a bit unexcitimg! The weather can be too hot in the height of summer, but for the rest of the year its wonderful for spending time outdoors camping, swimming, surfing, hiking, biking etc. etc.
The MTB scene is small compared to the UK, but growing very rapidly, and I've been hitting the trails every weekend for a year now, and still haven't got bored!
The lifestyle is typical Ozzie suburban (think Neighbours!) and can be a bit parochial, but that all depends on whether you like to spend your time at the theatre & galleries, or at the beach & BBQ. The rest of Australia is a long flight away, but if you're on a good salary then this isn't a problem, and all the possibilities of Asia are on your doorstep.
The economy is currently booming in WA because of the resources industry, which has created a "two speed economy". Meaning that if you work in mining or related industries you'll be very well off, but for the rest of us its a bit of a double edged sword in that it forces rents, prices etc higher. Migrants in the medical profession used to have to do their time in rural communities before living in Perth, which is a whole different world, so better check how the current rules would apply to your wife.
Whether you'll settle well in Perth or not will probably depend on your attitude more than anything else. They may speak English, but its still a foreign country with its own culture and way of doing things. In many ways its more like suburban America than England, but without the poverty & crime! And don't forget you are the guest in their country - too many Brits come over with a superior attitude thinking they are 'owed' something, and quickly crash & burn.
If your life is at a natural crossroads, you're looking for a change and a fresh start, and you're willing to make the effort to fit in, then you should thrive. However, if you're leaving behind a very strong network of friends and family, then you may not survive the transition period where you have to build up a totally new support network from scratch. Don't worry about the kids, they're the most adaptable, its you and your wife who will need to think carefully whether you're really ready to start all over again, because thats the reality of moving half way around the world.
Hope those thoughts are of some help - I'm sure you'll get a variety of views from others on here...