Hi! Sorry this took so long, was out of internet range for a bit. As far as my personal knowledge goes, Didsbury might be your best bet if you do want to me near to the city centre. Manchester has over 30,000 students (between the two universities here), so they tend to be everywhere, but lots of the people from my work chose Didsbury because it's an easy commute (will you have a car?) by car or bus, and you can find some nice semi-detached properties. There are "some" students but not many, really.
It's not really walking distance to downtown (you'd have to look more at Rusholme, Fallowfield or the city centre for that and there would be more students!), but it's an easy bus ride, and there are some nice restaurants out there. I'd definitely try and work out your commute first thing, because the roads can be absolutely jammed during rush hour around here. Take whatever Google says and add at least 20 minutes--and that's just normal traffic

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You can find some Mexican stuff here! Lots of black beans, some peppers and etc. I still miss proper Mexican food, though. I'm starting to miss curry when I go visit the US though, so you can't win!
I'd be happy to talk to you about places to ride, though for that you really need a car (it was my reward for getting off my butt and getting my UK license!). There's LOTS to do in Manchester; it's a great place!