Commute - really depends on where you live, as others have said. I grew up on the North shore, which is a lovely green area, good for families - expensive housing. If you were to live there, not super close to the harbour (very, very expensive housing), but say 15 minutes drive, your DH would be looking at a 30-50minute commute to the city at peak hour, depending on how far you live from a station. If you want to look on Domain.com, some suburb names, to get you started: Chatswood, Lindfield, Roseville, ...and then a bit further still by car (I suspect roughly the same commute time?) : Ryde (is aso cheaper), Epping.
Posh suburbs near the harbour (Mosman, Neutral Bay, Cremorne) are gorgeous, and you can commute quicky if you live close to the ferry...but, thy have no trains, and the bus service in Sydney is definately slower than trains (because of traffic mainly).
I hope that helps. Others on here will be able to help with suburbs in the Inner West and South sides - not my specialty.
Regarding your sons education - have I understood correctly that he is not yet at school? If thats the case, and there is no worry about disrupting his current schooling, then really I wouldn't worry. As I said, the schools here are, on the whole, excellent. The main differences are that the starting age is slighly later (a good thing imho), and that the schools seem, to me, to be less "pushy" ain key stage one than UK schools are - ie, they are a bit more play focussed (also a good thing imho). Some of the local primary schools close to the city can be pretty crowded - if there is a high enrollment they still have to take all the kids, and so put demountable clasrooms in the playgrounds...often reducing play space to a poor leve. Lots of other schools, particularly those slightly further from the city (in the suburbs I mentioned fr example) there are wonderfully spacious schools, with ovals, playgrounds, bush, loads of room to run, only 200 or 300 pupils and very good educational standards.
What do we do in the evenings? Well, my kids are 2 and 4 and I am pg with number 3...so at the moment we stay in and I feel sick! Normally, we mght have people over for dinner (seems to happen a lot more in Sydney than where I lived in the UK), go to a friend's house, have a BBQ...in summer, go for a walk to local park with the kids...if we have a night out it would usually be to go to friend's houses for dinner, or to the movies, meybe out to dinner at a restaurant (cheaper here than UK), theatre...same sort of stuff as in the UK really, but more socialising in people's homes I think.
A normal day for us - well, we just got back from the UK 8 weeks ago and are very much still setting up here...but here is what I think life will be like as of next week:
DD (4.5) goes to Kindy (nursery) 3 days a week (9-3pm...and not free by the way - it costs us roughly £20 per day to send her), so we will get up at about 7.30am, leave at 8.30 to walk to kindy. I would do things like housework (not much!), laundry, take DS (3) to toddler groups or meet up with friends and kids in a local park, maybe sometimes take DS to the zoo or aquarium. Thursdays - DD not at kindy and both kids have a swimming lesson at a pool about 15 minutes drive away (one of 5 local pools we could have chosen!). Then we stay at the pool and have a swim/splash, driv home, kids faff about in the garden or play with toys.
DH (currently job hunting) I expect will get home at about 7pm ish..hopefullly earlier sometimes. Have dinner, kids to bed...
TBH, it is the same as what we did in the UK, except we spend much more time outdoors here. Even in winter we will be able t take the kids to the park, ride bikes, go for walks, walk to nursery etc...of course it rains sometimes, and sometimes is unpleasatly cold (relatively speaking!), but on the whole, even winter is mild.
I hope that helps - I feel very self obsessed writing 'a day in the lif of astrophe' like that...but you did ask!
Skiing - you can ski here! I'm sure it othing compared to the Alps, but it is popular here, although the season is fairly limited. New Zealand has great skiing (so I'm told) and is only a short flight away.
DH's holidays - 4 weeks a year is the standard, although often companies close down between Christmas and New year, so it may be more like 5. Obviously depends on the company.
Re public transport and driving - unless you lived right in Sydney city, you will need a car. It is a very sprawling city - very spread out. If your DH can catch public transport to work, and if you choose to send you DS to a local school, you will probably get away with not using a car too much, but it is pretty unlikely you wil be able to get everywhere you need to go n public transport...or if you can, its likely to take you a long time changing busses etc.