Ok, so I'm going to post exclusively about Manchester as I rarely venture into Trafford. I'm very much a South Manchester girl so my answers are biased in that direction, and I don't have a child (yet - there should be one this time next week!)
My favourite thing about Manchester (the city of) is how vibrant and diverse it is. There are plenty of shops, museums, galleries and libraries, the suburbs range from rough to fancy, the people are warm and chatty and the nightlife is brilliant. Every 2 years there is the Manchester International Festival which programmes new theatre, music & art. It has seem incredible events see here at www.mif.co.uk.
Also the Literature and Children's Literature festivals are worth exploring.
My least favourite thing? Lack of green space in the city centre, all the great cities have a park. Also, Mancs are a little bit smug about being from Manchester... I know I am.
When I have friends with children visiting I like to take them to the Lead Station in Chorlton, Manchester Art Gallery Cafe, The Buddhist Centre (Vegan) or Croma (City Centre, Chorlton or Prestwich). These all have yummy food at reasonable prices and amuse the children without being completely hideous.
For a child-free night out (or what I am still just about calling a 'night out') I like real ale and cocktails. So a decent pub crawl might start at the Marble Arch on Rochdale Rd, then The Angel. Into the Northern Quarter via The Castle (check out the amazingly unrestored ceiling) and onto maybe Matt & Phreds for caipirinhas and jazz, Odder and Trof. Finish with dirty cocktails at Alchemist and a curry at This & That cafe.
For outside space it's best to head away from the Town Centre, Heaton Park to the north of the city is huge, Wythenshawe park to the south is vibrant and under-celebrated. Bramhall park has lovely woodland walks and ducks, Cholton Water park has the friendliest running club in Manchester, a sensory garden and a jolly icecream van.
My favourite hidden gem has to be Aumbry, a small but wonderful fine dining restaurant in Prestwich. Costs me £60quid in a taxi to get there and it's worth it. The food is amazing and the staff are lovely. I really cannot recommend their tasting menu enough.
I am biased when it comes to visitor attractions as I trained here, but for me John Rylands Library is the most beautiful building in Manchester. Has a great kids programme, including screening of Doctor Who and sleepovers, and a lovely cafe.
I don't know anything about schools yet. I'd assume that Didsbury or Chorlton would be your best bet.
Finally, why move here. Well I suppose because you have a job here, or family. I never moved away (well I tried but nowhere else is as good so I kept on coming back)