Hello,
We have lived in Antwerp for about 6 months and love it!
We don't have children yet, I am heavily pregnant so by spring we will, but from what we can tell the facilities for children are very good.
We live in the city centre so I can't advise on the suburbs. Within 5 minutes walk from our apartment is a huge park with a jungle themed playground, a big UGC cinema which shows some films in English, a swimming pool and a lovely zoo. There are also lots of lovely child friendly cafes in the city and also lots of suitable chains, pizza hut, macdonalds, wagamama etc
The Brussels Childbirth Trust, is about a half hour/40 minute drive away and organises lots of groups and events for ex pat children and parents/parents to be, all in English.
Most shops are open everyday except for Sunday and Monday morning. There is a big market in the city centre every Sunday morning that sells literally everything and some small corner shops are also open so you can still get emergency supplies.
Like everywhere with the Euro some things are more expensive than at home. One thing that makes a big difference for us is that our rent is inclusive of all bills including utilities and taxes down to internet and cable tv. We would not get anywhere near the same value for money at home.
I hope this helps and am happy to answer any other questions you have.
ps forget corner shops, have just remembered Ikea is also open on a Sunday