I agree, avoid St Kilda if you can. CBD is central to everything and nice walks to south bank and through the city itself. Brunswick St is good for shopping and breakfast. I always love the Botanic gardens for a walk or picnic. There is a lovely childrens garden there too.
Abbotsford convent has nice market days, good food and right next door to the Collingwood Childrens farm.
There is a great website for things to do with kids in Melbourne. I'm not sure if I can give details here but PM me if you want details.