I’d recommend looking in the inner west. You can find some large terraces, spacious for your family (and perhaps all that hiking/sport as well as your luggage’s!) The prices are quite decent and you’ll super close to the train or the light rail which you get you around in no time.
As for the gym and keeping fit. I would highly recommend recommend reaching out to Ben Connor, I’ve training with him for years. He’s now based in Camperdown fitness gym which is near Pyrmont, Glebe and Newtown. He is offering free sessions atm, so whilst you settle in, if you have free time I’m sure he can fit you in to get you back into your routine.
You can reach him on 0428136914 or via email:
Feel free to check out his website as well: www.universalmovement.com.au
Hope that helps and welcome to Aus!!
Any suburbs north of harbour bridge or inner west or south towards Cronulla shire are considered good safe family areas.
Avoid western Sydney completely.