Welcome to the forum :) I somehow missed replying to you the other day, my bad.
Anyways, exciting times ahead. A 457 visa is a temp visa and often you'll hear expressed its better to try to get a PR visa if you can. I tend to agree with this sentiment, more as its more security for those moving over in terms of not being tied to the job or if the person sponsored lost their job having the stress of finding another sponsored one within a short timeframe to enable you all to stay in Aus.
Being tied to an employer can also have its downside if you really hate the job or there are issues with it for whatever reason. Also I have read on occasion the non sponsored partner finding it hard to secure work on a temp visa too. I'd say go for a 457 if you didn't have kids but as you do, it kind of does make me nervous for you. Having said that, the nursing route may be ok to go down on the 457 if you can gain PR asap should you wish to remain in Aus long term.
School wise it will really depend on where you end up working, how far you want to commute and where you settle for the longer term. Some move right down south, areas which also include towns in their own right and quite a ways from Adelaide itself if say down in Aldinga, others not quite so far, lots settle in the metropolitan area, some north, north east... Adelaide is a big sprawl and you could end up anywhere that suits you wants/needs. Personally, I love where we are, I guess south eastern hills region it is, but would also happily settle in a few of the outer eastern suburbs or more up in the hills further out so being a bit more rural. I also love Clarendon and a few small towns up in the hills. Take your time, look at realestate.com.au and put suburbs etc in to check out property in each one and see what the prices are like. Certain areas are going to cost a lot more to buy in than others though good priced rentals can be found but the state of the house may be dubious.
Childcare for the ages you say is easy IMHO. OSHC is the out of hours school care attached to most schools. They are usually well run, popular and start around 7am in the morning and go till 6pm ish Mon-Fri in term time and then all day during school holidays apart from a week or two around Christmas when they tend to close for a break. Also public holidays etc they close. Its not expensive and my son loves it there and the day trips they do in the holidays as well as the on site programme. Its good as he goes to his school one so kids he knows including a fair few of his mates are also booked in after school and over the holidays. They have a blast. Cost wise we pay about $20 for after school and about $50 for a full day of vacation care. We get a rebate and elect to have this in a lump sum at the end of the tax year but I think you can also opt to pay less over the year if you prefer. If we did this we'd pay $10 for a school session and about $25 for the day.
My suggestion would be to sit down and think about what sort of lifestyle you have now and if it suits you, in terms of do you live in a built up area or on the outskirts, rural or right in the middle of a city. Is living closer to the beaches and coastline important to you when you move here (fwiw we live more inland in the foothills and its a 10-15 minute drive in the car to get to any number of beaches as they are not a priority for us and we are happy just visiting whenever we feel like it). Do you want to be closer to the actual city for work or social reasons? Do you want to be more on the outskirts of it all? What do you like to do with your family time? Play sport, go hiking, days out, cycling? Cinema, shopping, theatre, make a list and see what you all come up with :)
Nursing wise I'm going to point you to our sister site, Poms in Oz an suggest you have a browse of the more recent posts in a few threads and read some replies to questions over there. Feel free to sign up over there if you've not already and post asking any nursing stuff if you don't get replies here (not sure we have many nurses active on here atm but there are one or two around).
There is this big nurses thread over there that might be of use to post to and ask questions
Also the search feature there gave me these threads when I put in 'nursing'
Job hunting wise, no clue but I'd figure they'd know over on PIO and be able to point you in the right direction to employment sites or nursing agencies etc.
Good luck with it all and hope to see you around here a bit more