While I don't live in Greater Manchester, my wife's gran lives in Knutsford and her brother and his fiance live in Withington (just in a bit from Didsbury) and DW did some job hunting in Cheshire and Lancashire a few years back.
Cheshire is definitely the 'Surrey of the North', lots of leafy towns and quaint villages and home of many a Man U. footballer. Meanwhile, Lancashire has the reputation of dirty, grungy towns and not being so nice on the whole, but I'm sure that's a blanket generalisation, although I found Bolton a bit grimy which is a bit harsh as we were only there a day or so.
One town that is out west of Manchester is Warrington which I think is on the Cheshire side barely but is heading out towards Liverpool and Blackpool. Anyone have any comments on Warrington?
Anything north of there is pretty much Lancashire. I've not heard great things about Wigan but maybe someone might be able to comment better.
And, of course, you have Chorley as in Chorley FM - "Coming in your ears"

Manchester is a really nice city but I suspect moreso for a couple than those with kids. Salford is the only city-within-a-city that I can think of. It is a city in its own right in the middle of Manchester. Unfortunately, it has an absolutely dire reputation but the canals and Lowery Centre bit are quite nice actually, I think they've worked to spruce it up as best they can.
Good luck on wherever you end up.
Matt