I'm in Port Kennedy which is right next to Warnbro. It is a nice area, in fact, I find it so much like a seaside village that it feels a lot like I am holiday a lot of the time.
We're only a 10 minute drive to the Warnbro train station and there are great, timely bus services from here. I used public transport exclusively for about 18 months and it was fantastic. The buses and trains are very clean, they run to time most of the time and they are not too expensive.
We're in a 4 x 2.5 x 2 and are currently paying $410 a week. It is in a small link road to a road running close to the beach and we can hear the ocean (and see glimpses from upstairs) when all is quiet at night - it is really nice.
The suburb itself is pretty safe. It has lovely roads without too much traffic and parks around what seems to be every corner! My older kids, who are 11 and 9, have been going to the local shops, playing at the park and meeting up with friends on their own for a couple of years now. They ride, walk or scooter to school at the local public primary and have never had any problems.
There are several public primary schools (strictly zoned like the rest of Perth), a catholic school and a lutheran school. There are many daycares, some of whom I have heard great things about, some not so great - I guess that's the norm though.
I didn't have my little one at Rockingham hospital, but I did have a lot of monitoring there and my DD also had her appendix removed there and I have to say, I haven't been able to fault them at all. We have also spent a fair amount of time at the emergency department and have been pretty impressed there too. Sure, you wait a long time sonetimes, but they have always been courteous and never dismissed my concerns.
The train commute to the CBD isn't too bad - check out the transperth site to look at train lines, their timetables etc and then work out what you would see as your maximum commute time then look at suburbs around there. The freeway isn't too far away from Port Kennedy - takes 15 minutes at the most to get there and being south of the river, it tends to be a lot less congested than north of the river.
In general though, Perth is a wonderful place for families. There are so many parks and activites that families can enjoy and it is a very pretty city. We moved here from NSW 3 years ago and haven't regretted it for a minute. The people are a lot more relaxed and friendly here but do be prepared for some things to take a bit of extra time as there is a very laid back attitude.
Aldi is something I miss, but in general, I have found my finances to not be too bad since we've been here despite paying more for groceries.
We're looking at moving in February as we wish to save for a house. We have seen an admittedly run down, but liveable 3 x 1 in Mandurah for $210 a week, so we're looking at heading down there so we can save for the next 4 years. We plan, however, to buy in the Rockingham region as it is just so nice. DH is FIFO so it doesn't really matter where we live ...
Good luck for your move - hopefully you will be happy here!