itsahen I disagree that you can blanket Whalley Range,Moss side, Hulme & fallowfield as 'Totally fine now'. Are you a landlord?
I wouldn't recommend any family move to Moss side, full stop.
The other areas above have had bad reputations going back for much longer than 15 years, and they haven't suddenly become better areas in the last 10-15 years. Yes there are pockets of improvement in Hulme but its mainly student/rentals. Fallowfield is mainly for Students. Whalley range may have pockets of reasonable streets to live on but you would have to choose carefully. All these areas have longstanding problems and crime issues - there are better places to live for the same money.
You can't really lump Chorlton in with that group, most of it is expensive and sought after, there maybe some parts on the Chorlton borders that may be a little rough/cheaper . Stretford has not changed a great deal in 30 years, I think its a safe place to live in general and many live there because of the good schools within Trafford. I've noticed more middle class type families moving to Stretford in the last few years as they have been priced out of areas like Chorlton/Didsbury/Heaton Mersey, Urmston etc.
Up to a 40 min commute requirement gives you A LOT of options, I'd stay away from the inner city areas and look further out where prices are more reasonable, family orientated and away from the crime hotspots.
Family friendly areas - I would suggest Sale, Timperley, Altrincham (All Metrolink) Urmston, Flixton, Cheadle, Edgeley, Heaton Chapel, Davenport.