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I used to live in those areas for years, albeit as a student/early working years - have been in the UK since 2010 (it's right by Sydney Uni, so lots of students around, although it has been getting increasingly pricey, so that was changing in favour of families). It was definitely becoming more family friendly as we moved over and there were lots of mums and prams around the cafes. There are lots of decent cafes, restaurants and bars around and a few shops. It's also really close to Broadway, which is a large shopping centre with a big supermarket, grocers, bottle shop , cinema etc... Not sure about play centres or kiddie friendly parks as wasn't really my focus at the time.
Transport wise there are plenty of buses into town from Newtown's main road - would take 20-30 mins in peak traffic, although to commute they are packed by the time they get to Newtown as it's near the end of the bus routes. I don't think Erko is as well served by buses but both are also on a train line (different lines) for a quick train into central (to change if you work up north) or direct onto the city circle which is perfect if DH will be working in town. Have a look at CityRail for times (and if you are used to the tube, take a deep, deep breath; public transport in Sydney is AWFUL in comparison, it won't matter where you live).
I adored the area but its biggest downfall is that it's not easy to get to the beach from. On public transport there was only 1 direct bus (to coo gee) but you could train to red fern and change to bondi jctn, then bus to bondi but with kids that would be hideous. Driving would be about 30-40 mins (traffic dependent) to the Eastern Suburbs beaches (Bondi, Coogee etc).
THe only other thing I'd add is be sure to check out the cost of living before you commit to rental prices. Food prices, even in the supermarkets, are really high compared to what you'll be used to coming from England.
Good luck. Sydney is a wonderful city and I adored my time there. The Aussies are a friendly bunch.
I am moving to Sydney from the uk with my husband and 5 month old baby for 6 months in January. We are trying to find somewhere to rent and have seen a flat in the erskineville/Newtown area. Just wondered if anyone could offer any advice about the area? I will not be working so looking for somewhere which is nice, has cafes for me to go to, good public transport to get into town and to beach and family friendly. We have been focusing our search on Coogee area as would have liked to be close to beach but struggling to find somewhere.