Well......being Manc and proud as I am, I'd say come to Manchester. Parts are rough yes, but not all of it but so is anywhere.
We have some amazing museums and a lot of them are free to get in, libraries, theatres, it is very culturally diverse....the night life is everything you could want...it can be as crazy as you like or as cool.
Now, I probably wouldn't live in Rochdale though....it's a fair trek outside the city centre plus it is quite rough in parts. If you were looking for somwhere outside Manchester that is cheaper to live, I would suggest somewhere like Bury...it's just been through an urban renewal so has a fantastic town centre with great shops, bars etc but is a 20 minute tram ride from Manchester but still pretty cheap to rent there. Somewhere else quite cool is Chorlton, it's quite studenty but the nightlife there is cool. It's almost like being in another country, lots of little bars and restaurants where you can sit outside, listen to jazz, funk, pretty much anything and it's also really pretty....
Having said all that, Leeds is pretty cool too....just the wrong side of the Pennines for me
Plus if you come to Manchester you HAVE to become a lifelong Manchester United fan