Didsbury is fine for when you are young, and its ok when you just start a family but its young and its more 'going out' orientated than some of the other parts of South Manchester. Once you hit your mid thirties you'll start to feel it. Its also relatively expensive for the NW. Prices in Didsbury have shot up in recent years. I know a number of people who have moved out after starting a family, because its both expensive and not necessarily for older families.
Altrincham now has a really good trendy market area, so its in someways the 'next step' if you are into that lifestyle and have kids. The schools have a particularly good reputation. Hale is its more upmarket neighbour and Sale has nice patches and good schools, but also bits that are not quite so great - but can be slightly cheaper.
Slightly further out from Didsbury you have Cheadle and Cheadle Hulme. It tends to be a bit more 'grown up' and a bit more suburbia. Its nice, but not necessarily everyone's cup of tea. I think parts 'lack a bit of soul' for want of a better phrase.
Further out still you have Wilmslow or even Knutsford. They both have trains into the city but are much more rural feeling. They are both small towns with their own character. You'll find more country style pubs out this way.
Most places in that band along the M56 between Cheadle and Knutsford and onwards as far as the Grappenhall area of Warrington are relatively nice. Some are more pricey than others, and even with a South East budget for a house, you can find it going less far than you might have anticipated. There are a large number of people who have moved up from the South East in the last decade or so. Its a lot more like being close to London than it used to be.
And as such, you should be able to find an area out of that lot which suits your taste. The only thing you should be aware of, is that public transport, whilst good in places, still leaves a lot to be desired at times. The lifestyle is still a lot more dependant on cars. Particularly if you are not going into the city centre.
I would also warn that the city centre is not necessarily the nicest of places at the moment. I love the city but the homeless and spice problem has got a lot worse in the last 18months. There are a couple of places that I would have gone a few years ago, which I'm becoming a lot less happy to be in - particularly in the evening. This is something I've never previously had an issue with or thought much about and always felt very safe.
I'd argue that if you want to get an idea of what its like to live in the area, staying in the city is not particularly reflective of life either. It has the nightlife and cultural centres but its more what you do if you are visiting. If you are coming to get a feel for living in Manchester, stay further out and drive - its the only way you'll get around more than one area anyway, otherwise you'll be more restricted in what you can do - and you can still do the city centre too if you really want.