Family with 2 yrs old child near Melbourne CBD

We are moving from Kiev, Ukraine to Melbourne on Feb 15, 2017. My wife is not very fluent in English and I'd like to be able to get to Melbourne CBD by bicycle or PT. Ocean is not a priority at all. Looking for places where you can walk with a child and find other children there to socialize with. 1 bedroom apartment would be just fine. As we've lived in a big city for the last 8 yrs, I think it will be boring to live in a quiet suburb where nothing really happens. Nightlife is not an option either. We don't want to buy a car. It would be great if a grocery store like Aldi was around the corner. 400$ pw is an absolute top we can afford right now as I don't have a permanent job in Melbourne right now. Does Richmond, South Melbourne and North Melbourne match the criteria mentioned? Is it allowed in Melbourne to ride a bicycle with a small child behind in a chair?