Another vote for Didsbury, but if you really need to be within distance of Picadilly station every day then I’d consider living in the city centre.
Somewhere like Castlefield in the centre would be nice, along by the canal. There’s loads of luxury flats there for dirt cheap rents (because they have built far too many). You could use that as a base and then move out of the centre when you know the area a bit better.
If you lived in Didsbury you’d have to be pretty hardcore to brave the Manchester traffic everyday rain or shine and it’s going to add at least 30-40 mins each way to your day to ride in to Picadilly (or to get the bus if it’s raining). There’s no direct off road bike route to the city centre form Didsbury either – well not that I’ve found anyway.
Other than that you could live out a bit in say Hale, Cheadle Hulme or Heaton Moor and then get the train to Picadilly every day?
Avoid Salford, Beswick, Ancoats, Moss Side, Hulme, Longsight, Levenshulme, Wythenshawe, Fallowfield, or anywhere North of the City Centre – unless of course you like getting mugged 😯