Don't live there anymore but born and bred in S Manc... There's lots of good primary schools in Manc - problems are not quite the same as in London but you will need to look them up.
Chorlton and Didsbury V central and accessible and still hip and still expensive for inner suburbia... (350k for 3 bed terrace etc)
Much of Stockport more suburban but more for your money - Heaton Moor nicest of the Heatons but most like Didsbury in style and price (not quite as expensive Victorian houses)
Bramhall, Wilmslow, Prestbury, Altrincham, Hale/Hale barns etc - more suburban and WAG-ish, less hip and multicultural, more space, considerably more expensive in many parts... Altrincham has advantage of metro though good for commuting
All close to M60 links for commuting
Have a property (MIL lives in it) in Ellenbrook which is a newish estate in Worsley with a good primary school - very desirable for young families. Very accessible for Leigh but is a newbuild and bigger houses there well in your price range I think but not so much for your money. Worsley and Roe Green villages more 'old' victorian properties and more local facilities and bits are more upmarket. All very close to M60 M61 links to west/north and quick into central Manchester
Walkden though close to some pretty rough areas. Mind you every where in Manc is close to some rough areas like any big city...
You might also consider, since you need to be West... the Warrington bits eg Appleton etc, some of which are quite nice or even some of the North Cheshire villages on the other side of the M6, And of the inner city north bits of Manc, Prestwich is considered the 'poshest' but I'd never live there! (sorry N Mancs...) but there are bits of Bolton that are pretty nice too...
HTH